The mechanical principles of engineering and architecture

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MECHASICAL PRINCIPLES

ENGINEERING

ARCHITECTURE.

HENRY MOSELEY, M.A.'F.R.S.

Ssaond American frtim Second Lcndtm Editi:

D. H. iIA.HAN, LL.D.

WITH ILLUSTRATIONS OK WOOD.

NEW YORK:

JOHN WILEY k SON, PUBLISHERS,

2 Clinton Hall, Astor Place.

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)r the SoutterB Iti«lrW

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EDITOR'S PREFACE.

The high place that Professor Moseley occupies in the
scientific world, as an original inyestigator, and the clear-
nees and elegance of the methods he has employed in this
work have made it a standard text hook on the eubjecte it
treats of. In undertaking its revision for the press, at the
request of the puhliehers of this edition, it has heen deemed
advisable, in view of the class of students into whose hands
it may fall, to make some slight addition to the original.
Tiiia has been done in the way of Notes thrown into an
Appendix, the matter of which has heen gathered fi-om
various authorities ; hut chiefly from notes taken by the
editor, whilst a pupil at the French militaiy school at Metz,
of lectures delivered by General Poncelet, at that time, 1829,
professor in that school. It is a source of great pleasure to
the editor to have this opportunity of pubhcly acknowledg-
ing his obligations to the teachings of this eminent soman,
who is distinguished not more for his high scientific attain-
ment, and the advancement he has given to mechanical
science, than for having brought tliese to minister to the
wants of the industnal classes, the intelligent success of
whose operations depends so much upon mechanical science,
hy presenting it in a form to render it attainable by the most
ordinary capacities.

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IV ebitoe's pbttace.

Tlie editor would remiirk that he has carefully refrained
ffom mailing any alterations in the text revised, except cor-
rections of typographical errora, and in one instance where,
from a repetition of apparently one of these, he apprehended
Eome difficulty might he offered to the student if allowed
to remain exactly as printed in the onginal.

Unitbd States Militabt Acadeut,

West Faint March 8, 1856.

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PREFACE TO THE SECOND EDITION.

I HAVE added in this Edition articles ■. — first, " On tlie
Dynamical Stability of Floating Bodies ;" secondly, " On
the EoUing of a Cylinder ;" thirdly, " On the descent of a
body upon an inclined plane, when subjected to variations of
temperature, which would otherwise rest upon it ;" fourthly,
'* On the state bordering upon motion of a body moveable
about a cylindrical axis of finite dimensions, when acted
upon by any number of pressures."

The conditions of the dynamical stability of floating
bodies include those of the rolling and pitching motion of
ships. The discussion of the rolling motion of a cylinder
includes that of the rocking motion to which a locomotive
engine is subject, when its driving wheels are falsely
balanced, and that of the slip of the wheel due to the same
cause. The descent of a body npon an inclined plane
■when subjected to variations in temperature, which other-
wise would rest upon it, appears to explain satisfactorily the

The numerous corrections made in the text, I owe chiefly
■jo my old pupils at King's College, to whom the lectures
af wliich it contains the substance, wei'o addressed. For

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VI PEEFACE 'ro THE 8EC0ND EDITION.

several important ones I am, however, indebted to Mr,
"Robinson, Master of tlie School for Shipwrights' Apprentices,
in Her Majesty's Dockyard, Portaea ; to whom I have also tc
express my warm acknowledgments for the care with
wliich he has coiTected the proof sheets whilst going through
the press.

Vay, 18»

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PREFACE.

In the following work, I have proposed to myself to apply
the piinciplee of mechanics to the discussion of the most
irapoi-tant and obvious of those questions which present
themselves in the practice of the engineer and the architect;
and I have sought to include in that discussion all the^
circumstances on which the practical solution of such queS'
tions may be assumed to depend. It includes the substance
of a course of lectures delivered to the students of King^S-
College in the department of engineering and architecture,,
during the yeai-s 1840, 1841, 1842 *

In the first pai-t I have treated of those portions of the
science of Statics, which have their application in the theory
of machines and the theory of construction.

In the second, of the science of Dynamics, and, under this
head,*particular!y of that nnion of a continued pressui-e with
a continued motion which has received from English wiiters
the various names of " dynamical effect," " efficiency," " work
done," " labouring force," " work," &c. ; and " moment
d'activit^," " quantite d'action," " puissance mecanique,"'
" travail," from French writers.

Among tlie latter this variety of terms has at length given
place to the most intelligible and the simplest of them,.

• The fivist 170 pages of Ihe work were printeil for the use of inj pupils in the
year 1840. Copies of tliem were about tte same time in tlie possession of
BCTeml of mj' frieada m the Univeraities.

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"travail." The English word "wort" is Hie obvious trans-
lation of " travail," and the use of it appeai-s to be recom-
mended by tlie same considerations. The work of overcoming
a pressure of one pound through a space of one foot has, in
this country, been taken as the unit, in terms of which any
other amount of work is estimated ; and in France, the work
of overcoming a pressure of one kilogramme through a space
of one metre. M. Dnpin has proposed the application of the
term dyname to this unit.

I have gladly sheltered myself from the charge of having
contributed to increase the vocabulary of scientific words,
by aeeuming the obvious term " unit of work " to represent
concisely and conveniently enough the idea which is attached
to it.

The work of any pressure operating through any space is
evidently measured in terms of such units, by multiplying
the number of pounds in the pressure by the number of feet
in tlie space, if the direction of tlie pressure be continually
that in which the space is desci-ibed. If not, it follows, by
a simple geometrical deduction, tliat it ie measured by the
product of the number of pounds in the pressure, by the
number of feet in the projection of the space described,*
upon the direction of the pressure ; that is, by the product
of the pressure by its virtual velocity. Thus, then, we
conclude at once, by the principle of virtual velocities, that
if a machine work under a constant equilibrium of the
]jressures applied to it, or if it work uniformly, then is the
aggregate work of those pressures which tend to accelerate
its motion equal to the aggregate work of those which tend
to retard it ; and, by the principle of vis viva, tliat if the
machine do not work under an equilibrium of tlie forces
impressed upon it, then is the aggregate work of tliose which
tend to accelerate the motion of the machine greater or less

" If the direction of the pressure remain always piiriillel to itself, tie space
fleacribed may be any finite apace ; if it do not, the spooo is tinderstood to be
eo small, that the direction of the pressure may be supposed to remain parallel
lo itself whilst that space is described.

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tlian the aggregate work of those which tend to retard ita
motion by one half the aggi-egate of the vires vwce acquired
or lost by the moving parte of the eystem, whilst the work is
being done npon it. In no respect have the labonra of the
illustrioua president of the Academy of Sciences more con-
tributed to tlie development of the theory of machines than
in the application which he has so successfully made to it of
this principle of ms tma.* In the elementaiy discussion of
this principle, which is given by M. Poncelet, in the intro-
duction to his Mecaniqi/s IndMst/rielle, he has revired the
term iiis mertice {vis inertiw, vis insita, Kewton), and,
associating with it the definitive idea of a force of resistance
opposed to the acceleration or the retardation of a body's
motion, he has shown (Ai-ts. 66. and 122.) the work expended
in overcoming this resistance through any space, to be
measured by one half the vis viva accumulated through the
space; so tliat throwinginto the consideration of the forces
under which a machine works, the vires inertim of its moving
elements, and observing that one half of their aggregate via
viva is equal to the aggi-egate work of their vires inerticB, it
follows, hy the principle of vu-tnal velocities, that the differ-
ence between tlie aggregate work of those forces impressed
upon a machine, which tend to accelerate its motion, and
the aggregate work of those which tend to retard the motion,
is equal to the aggregate work of the vires inertice of the
moving parts of the machine : under which foiin the prin-
ciple of vis vi/oa resolves itself into the principle of virtual
velocities. So many dilficulties, however, oppose themselves
to the introduction of the term vis mertim^ associated with
the definitive idea of a force, into the discussion of questions
of mechanics, and especially of practical and elementary
mechanics, that I have thought it desii-able to avoid it. It
is with this view that I have given a new interpretation to
that function of the velocity of a moving body which is
known as its vis viva. One half tliat function I have inter-
preted to represent the number of unite of work aoctimulated

*See Poaeelet, Mecaniqiie [ndaslrielle, troisierae partie.

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in tlie body so long as its motion is continued. This numlDei
of units of work it is capable of reproducing upon any resist-
ance opposed to its motion. A very simple investigation
(Art. 66.) establishes the truth of this interpretation, and
gives to the principle of vis vma the following more simple
enunciation : — " The difference between the aggregate work
done upon the machuie, dra-ing any time, by those forces
which tend to accelerate the motion, and the aggregate
work, during the same time, of tliose which tend to retard
the motion, is equal to tlio aggi-egate number of units of
work accumulated in the moving parts of the machine
during that time if the former aggi'egate exceed tlie latter,
and lost from them duiing that time if the foraier aggregate
fall short of the latter." Thus, then, if the aggregate work
of the forces which tend to accelerate the motion of a
machine exceeds that of the forces wliieh tend to retard it,
then is the surplus work (that done upon the driving points,
above that expended upon the prejudicial resistances and
upon tiie workiug points) continually accumulated in the
moving elements of tlie machine, and their motion is thereby
continually accelerated. And if die former aggi'egate be
less than the latter, then is the deficiency supplied fi'om the
work already accumulated in the moving elements, so that
their motion is in this case continually retarded.

The moving power divides itself whilst it operates in a
machine, first, into that wliich overcomes the prejudicial
resistances of the machine, or those which are opposed by
friction and other causes, uselessly absorbing the work in its
transmi^ion. Secondly, into that which accelerates the
motion of the various moving parts of the machine, and which
accumulates in them so long as tlie work done by tiie moving
power upon it exceeds that expended upon the various
resistances opposed to tiie motion of the macliine, Tliirdly,
into tiiat which overcomes tiie useful resistances, or tliose
which are opposed to the motion of the macliine at the
working point, or points, by the useful work which is dona
by it.

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Between these three elemeiita there obtains in every
maohine a mathematical relation, which I have called its
MODULUS, The general form of this modulus I have diaciiased
in a memoir on the " Theory of Machines " prihlished in the
PhUosoph/ical Transaotiona for the year 1841. The deter-
mination of the particular moduli of those elements of
machinery which are most commonly in use, is the subject
of the third part of the following work. From a combination
of the moduli of any such elements tliere results at once the
modulus of the machine compounded of them.

When a machine has acquired a state of uniform motion,
work ceases to accumulate in its moving elemente, and its
modulus assumes the form of a direct relation between tlie
work done by the motive power upon its driving point and
that yielded at its working points, I have determined by a
general method* the modulus in this caae, from that statical
relation between the driving and working pressures upon
the machine which obtains in the state bordering npon its
motion, and which may be deduced from the known condi-
tions of equilibrium and the established laws of friction. In
making this deduction I have, in every case, availed myself
of the following principle, first published in my paper on the
theoiy of the arch, read before the Cambridge Philosophical
Society in Dec. 1833, and printed in tlieir Trwnsactions of
the following year: — "In the state bordering upon motion
of one body upon the surface of another, the resultant
pressure upon their common surface of contact is inclined
to the normal, at an angle whose tangent is equal to the
coefficient of friction."

Tills angle I have called the limiting angle of resistance.
Its values calculated, in respect to a gi-eat variety of sm-faces
of contact, are given in a table at the conclusion of tlie
second part, from the admirable experiments' of M. Morin,"!-
into the mechanical details of which precautions liave been
introduced hitherto unknown to experiments of this class,

■Art. 162. See PMl. Trans., 1841, p. 290.

\ Nonveltes Experiences sur Is FroUemerti, Paris, I8S3.

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and wliich have given to our knowledge of the laws of
friction a pi'eciaion and a certainty hitherto unhoped for.

Of the various elements of machinery those which rotate
ahout cylindrical axes are of the most frequent occun-ence
and the most useful application; I have, therefore, in the
first place sought to estahlish the general relation of the
state bordering upon motion between the driving and the
■working pressures upon such a macHne, reference beiag
had to the weight of the machine.* This relation points out
tihe existence of a particular direction in which the driving
pressure should be applied to any such machine, that the
amount of work expended upon the friction of the axis may
be the least possible. This direction of the driving pressure
always presents itself on tlie same side of the axis with that
of the working pressure, and when the latter is vertical it
becomes parallel to it ; a principle of the economy of power
in macliineiy which has received its application in the
parallel motion of the marine engines known as the Gorgon
Engines.

I have devoted a considerable space in this portion of my
work to the determination of tlie modulus of a system of
toothed wheels ; this determination I have, moreover,
extended to bevil wheels, and have included in it, with the
influence of the friction of the teeth of the wheels, that of
their axes artd their weights. An approximate form of this
modulus applies to any shape of the teeth under which they
may be made to work correctly ; and when in this approxi-
mate foiTn of the modulus the terms which represent the
influence of the friction of the axis and the weight of the
wheel are neglected, it resolves itself into a well known
theorem of M. Poncelet, reproduced by M. Navier and the
Eev, Dr. "WhewelLf In respect to wheels having epicy-

" In mj memoir on the " Theory of MachiBes " {Phil 7'rans. 3641), I have
extended this relation to the case in nhich the cumber of the pressures aad
their directions are any whatever. The theorem which expresses it is given in
the Appendix of this worlt.

\ In the disousMOn of the friction of the teeth of wheels, the direction of the
mutmit pressures of the teeth ia determined by a method first applied by me to

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cloidal and involute teeth, the modulus asaames a character
of mathernatieal exactitude and precision, and at once
establishes the conclusion (so often disputed) that the loss of
power is greater before the teeth pass the line of centres
than ai corresjponding pomts afterwards ; that the contact
should, nerei'theless, in all cases take place pai'tly before
and partly after the line of eentfes has been passed. In the
case of involute teeth, the proportion in which the arc of
contact should thus be divided by the line of centres is
determined by a simple formula ; as also are the best
dimensions of the base of the involute, with a view to the
moat perfect economy of power in the working of the
wheels.

The greater portion of the discussions in the tliird part of
my work I believe to be new to science. In the fourth part
I have treated of " the theory of the stability of stnictures,"
refen'ing its conditions, so far as they are dependent upon
the rotation of the parte of a structure upon one another, to
the properties of a certain line which may be conceived to
ti-averse every structure, passing tlu-ough those points in it
whore its surfaces of contact are intersected by the resultant
pressures npon them. To this line, whose properties I first
diecnsse^ in a memoir npon " the Stability of a System of
Bodies in Contact," printed in the sixth volume of the Comb.
Phil, Trams., I have given the name of the line of resists
ance; it diffei-s essentially ia its properties from a line
referred to by preceding writers under the name of the
curve of equilibrium or the line of pressnre.

The distance of the line of rtsistance from the extrados of
a structure, at the point where it most nearly approaches it,
I have taken as a measure of the stability of a structure,* and

that purpose in n popular treatise, entitled Mechanics applied to ike ArU,
publiahed in X834.

" This idea was suggested to me by a rule for the stability of revetemeiit
walls attributed to Vauban, to the effuct, that the resnltaut prsssare should
ioteraeot the baae of such a wall at a point whose distance from its eitrados is
i.ths the distance between the extrados at the base and the vertical through
the centre of gravity.

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have called it the modulus of stability ; conceiving thia
measure of the stability to be of more obvious and easier
application than the coefScient of stability used by the
Fpench writei"s.

That structure in respect to every independent element
of which the modulus of stability is the same, is evidently
th<i structure of the greatest stability having a given quantity
of material employed in its construction ; or of the greatest
economy of material having a given stability.

The application of these principles of construction to the
theory of piers, walls supported by counterforts and shores,
I walla supporting the thrust of roofa, and the
i of the floors of dwellings, and Gothic structures,
has suggested to me a class of problems never, I beUeve,
before treated mathematically.

I have applied the well known principle of Coulomb to
the determination of the pressure of earth upon revStement
walls, and a modification of that principle, suggested by jVE.
Poncelet, to the determination of tlie resistance opposed to
the overthrow of a wall backed by earth. This determina-
tion has an obvious application to the theory of foundations.

In the application of the principle of Coulomb I have
availed myself, with great advantage, of the properties of
the limiting angle of resistance. Ail my results Have thus
received a new and a simplified form.

The theory of the arch I have discussed upon principles
first laid down in my memoir on " the Theory of the Stability
of a System of Bodies in Contact," before referred to, and
subsequently in a memoir printed in the "Treatise on
Bridges" by Professor Hosking and IMj. Hann.* They
differ essentially from those on which the theory of Coulomb
is founded ;f when, nevertheless, applied to the case treated

* I hare made estendve use of the memoir above referred to in the following
■work, by the obliging perrois^on of tlie publisher, Mr. Weale.

I The theory of Coulomb was unknown to me at the time of the puhlioatiou
of my memoirs printed in the Camh. Phil. Uraits, For a comparison of the
two methods see Mr. Hann's treatise.

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by tbe French mathematiciana, they lead to identical results.
I have inserted at the conclusion of my work the tables of
the thrust of circular arches, calculated by M. G-aridel from
formulfe founded on the theory of Ooulomb.

The fifth part of the work treats of the "strength of
materials," and applies a new method to the determination
of the deflexion of a beam under g^ven pressures.

In the case of a beam loaded uniformly over its whole
length, and supported at four different points, I have deter-
mined the several pressures upon the points of support by a
method applied by M. Navier to a similar determination in
respect to a beam loaded at given points,*

In treating of rapture by elongation I have been led to a
discussion of the theory of the suspension bridge. This
CLTiestion, so complicated when reference is had to the weiglit
of the roadway and the weighte of the suspending rods, and
when the suspending chains are assumed to be of uniform
thickness, becomes comparatively easy when the section of
the chain is assumed so to vary ite dimensions as to be every
where of the same strength. A suspension bridge thua
constructed is obviously that which, being of a given
strength, can be constructed with the least quantity of
materials ; or, which is of the greatest strength having a
given quantity of materials used in its const ructi on. f

The theory of rupture by transvei^e strain has suggested
a new class of problems, having reference to the foiins of
girders having wide flanges connected by slender ribs or by
open frame work : the consideration of their strongest forms
leads to results of practical importance.

In discussing the conditions of the strength of breast-
summers, my attention has been directed to the best positions
of the cohimns destined to support them, and to a comparison

• Ae JQ ig. p. 487. of the following work.

I That partiouiar case of tliis problem, in whioh the weights of the suspending
rode are neglected, has bean tronted by Mr. Hodgkmson in the fouctli vol. of
Maitchetter ID'ansaations, with his usual ability. He has not, howeyer, suc'
Ceeded in effecting ita complete solution-

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of the sti'engtli of a beam carrying a uniform load and sup-
ported freely at its extremities, with that of a beam similarly
loaded but having ita extremities finnly imbedded in
masonry.

In treating of the strength of columns I have gladly
replaced the mafliematical speculations upon this subject,
which are so obviouelj 'founded upon false data, by the
invfduable experimental results of Mr, E. Hodgkinson,
detailed in his ■well known paper in the PhUosqpMcal
TrcmsacUons for 1840.

The sixth and last part of my work treats on " impact ;"
and the Appendix includes, together with tables of the
mechanical properties of the materials of construction, the
angles of rupture and the thrusts of arches, and complete
elliptic functions, a demonstration of the admirable theorem
of M. Poncelet for determining an approximate value of the
square root of the sum or difference of two squares.

In respect to the following articles of my work I have tc
acknowledge my obligations to the work of M. Poncelet,
entitled MScamque Industridle. The mode of demonstration
is in some, perhaps, so far varied as that their origin might
with difficulty be traced; the principle, however, of each
demonstration— flll that constitutes its novelty or ita value —
belongs to that distinguished author.

30,* 88, 40, 45, 46, 47, 52, 58, 62, 75, IDS,! 123, 202,
267,$ 268, 268, 270, 349, 354, 365.§

* The enuQoiatio!! ooly of this theorem ia ^veu in the Jfic. iiiA, 2rae partie,
An. 38.

\ Some importajit elementa of the demonatration of this theoL'em are taten
from the Mec. hid., Art. 79. Sine partie. The principle of tlie demonstration
ia not, however, the aame aa in that work.

X la this and the three following articles I have developed the theory of the
fly-wheel, under a different foim from that adopted by M. Poncelet (Mic. Ind.,
\vt. 68. 8me partie). The principle of the whole oaloulaUon is, liowever,
taken from his work. It probably consatutea one of the most valuable of his
contributiona to pi'ac^cal science.

§The idea of detfirmhiing the work necessary to produce a given deflection
of a beam from that expended the compression and tlie elongation of ita com-
ponent fibres was suggested by an oliservatioii in the Men. Ind., Art. To. 3me

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OOHTEWTS.

The Parallelogram of PresBures
The Principle of the Equalitj of 11

The Polygon of Pressures 10

The Parallel opipedon of Pressures 14

Of Parallel Pressures ^ ..... IS

The Centre of Gravity . . = . 30

The Properties of Guldious ..,.,.,,. 88

Motion 4,1

Velocity 43

Wore 48

Work of Pressures applied in different Direetiona to a Body moTeable

about a fixed Asia , 6T

Accumulation of Work . . . 63

Angular Velocity . . 6S

The Momect of Inertia 70

The Acoeleration of Motion bt oiteh motikq Forces . . .79

The Desoent of a Body upon a CarTB 83

The Simple Pendulum Sfi

Impulsive Force 86

The Parallelogram of Motion 88

The Polygon of Motion 88

The Principle of D'Alembert , ... 89

MoUau of Transkljon , , . . 90

Motion of EotatioQ about a fiied Axis 91

The Centre of Percussion 96

The Centre of Ofioillation 96

Projectiles . s . .99

Centrifugal Force ~ ... 106

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TliB Priuoiple of virtual Velocities 112

The Principle of Via Yiva 115

Dynamical Stability ' , 121

FaicnoB 124

Summary of the Laws of Friction 130

The limilJDg Angle of Ke^stancG 1">1

The Cone of Redstance 133

The two States botdoring upon Motion IBS

Tug aiftLDitY OP Conns 142

PAKT III.

The Tranemiasion of Work by Machlnaa . 1*6

Tlie Modulua of a Machine moving with a uniform or periodical Motion . 149
The Modulua of a Maoliine moving with an accelerated or a retarded

Motion 150

The Telocity of a Machine moving with a variable Motion . . .161
To determine the Co-efficients of the Modulua of a Macliine . . . 16S
General Condition of the Stat« bordering upon Motion in a Body acted
upon by Pressures in the same Plojie, and moveable about a cylindrical

Axis 154

The Wheel and Axle 15S

The Pulley ISO

Syetem of one fised and cue moveable Pulley IGl

A Sjstem of one fixed and any Sumber of moveable Pulleys . . .US

A TacMe of any Number of Sheaves 166

The Modulua of a compound Machine 1S9

The Capstan "134

The Chinese Capstan 199

The Horse Capstan, or the Whim Gin 203

The Friction of Cords 207

The Frioljon Break ai3

The Bmd 215

The modulua of the Band , . . .217

Tlie Teeth of Wheels 227

Involute Teeth 2Si

Epicycloidal and Hjpoeydoidal Teeth 2SB

To set oat the Teeth of Wheels 2Bfi

A Train of Wheels 241

The Strength of Teeth 24,1

To describe Epicycloidal Teeth 245

To describe involute Teolh 251

The Teeth of a Bacli and Pinion ........ 353

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The Teeth of a Wheel working with a LtintcrQ or Trundle . . . ae'i

The driving and worlting Pressm-es on Spur Wheels 259

The Moauiua of a System of two Spur Wheeis 268

The Modulus of a Hack and Pinion 283

Conical or Bevil Wheels 284

The Modulus of a System of two BeTil Wheels 288

The Modulus of a Tridn of Wheels 30!

The Train of least Reaiataiice 31l>

The iDoliued Plane 312

The Wedge driven by Pressure 321

The Wedge driyea by Impact 323

Tbo mean Preeaure of Impact S2.n

The Screw 826

Applications of the Screvr S2S

The Differential Screw ......,,.. E31

Hunter's Screw 333

The Theory of the Screw with a Square Thread in reference to the vari-
able Indumtion of the Thread at different Distances from the Axis . 333

The Beam of the Steam Engine SS7

The Crank 341

The Dr.ad Points in the Crank 345

The Double Crank 348

Tiie Crank Guide 851

The Fly-wheel 353

The Friction of the Fly-wheel 36-2

The Modulus of the Crank and Fly-wheel 363

The GoTernop 3(14

The Carriage-wheel S6S

On the State of the accelerated or retarded Motion of a Miwhine . . 313

General Conditions of the Stability of a Structure of Dncemeiited Stones R77

The Line of Eeeiatanee , . • . 377

The IJne of Pressure 87^1

The Stability of a Solid Body 3S!)

The Stability of a Structure 381

The Wall or Pier 8S'J

The Line of Resiataoce in a Pier 383

The Stability of a Wall supported by Shores 387

The Gothic EuttreM 336

The Stability of Walls sustaining Itoofs 3m

The Plate Band 403,

The sloping Buttress lOJ

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Tlie Stability of n Wall sustaining the Prtsauro of a Fluid , , ,403

Earlh Woriis , 412

Heveteniftit Walla IIB

The Arch 4^3

The Angle of Rupture 431

The Line of R^sistniice in a circular arch whosa Vousaoirs are equal, and

■whose Loftd is dietribnted oyer different Points of lis Extrados . . 440

A segmental Arch whose Eittrados k horizontal 441

A Oolliic Ai'ch, Che Estradas of each Semi-Arch being a. straight Lino

inclined at any ^ten Angle to the Hovizon, and the Material of the

LfladiQg different ft'om that of the Areh 442

A circulnp Areh having equal Vouasoire ajid Euataiuiiig the Pressare of

Water 444

The Equilibrium of an Arch, the Contact of whoso Voussoira ia geometri-

Applicationa of the Theory of the Avoh 448

Tables of the Tliraat of Arches 454

Elasticity ............. 433

Elongation 459

The Moduli of Reahenoe and Fragility 462

Deflection 4G7

The Defleidon of Beams londod uniformly 4B1

The Defleiion of Breast Summers 489

Rupture 502

Tenacity 602

The Suspen^on Bridge 505

The Catenary 5i)B

The Suspension Bridge of greatest Strength 610

Rupture by Compression S18

The SeotJon of Rupture in a Beam 62il

Gflnetal Conditions of the Rupture of a Beam 62!

The Beam of greatest Strength . . . . . . . . 527

The Strength of Breast Summers 54i)

The beat Positions of their Points of Support B42

Fomiulfe representing the absolute Strength of a Cyiindrioal Column to

austoin a Pressure in the Dicection of its Length 54S

Torsion 54e

, Google

The Impaot of two Bodiaa wKcse centres of Gravi

ffreatest Compression of tbe Surfaca of tho Bofllee .

Velocity of two elastic Bodies after Impact

The Pile Driver

ADDWIOHS BT the AMEEICiH EDITOR

APPENDIX.

KoteA 931

Note B.— Ponoelet's Theorema BSa

Sole 0.— On the Rolling of Ships 637

Note D 653

Note K— On the Rolling Motion of a Cylinder 6SS

Sote F. — On the Descent upon an Inclined Plane of a Boily subject to
Variations of Temperature, and on the Motion of Glaciers . . .6TB

Note 0,~Tlie best Dimensions of a Buttreaa 688

Note H.— Dimensiona of the Teeth of Wheels 684

Note I. — Experiments of M. Moria on the Traction of Carriages . . 885

ffote K.— On the Strength of Columna 686

Table I. — The NumBrieal Values of complete Elhptie Fimetions of the
Jirat and senond Orders for Yalnea of the Modulus fc corresponding to

each Degree of the Angle lin.-'k 6B7

Table II.— Shovfing the Angle of Buptnre t of an Arch whose Loading
is of the same Material with its Vouaaoirs, and whose Extrados is

inclined at a giren Angle to the Horizon 688

Table III.— Showing the Horizontal Thrust of an Arch, the Radius of
whose Intcados is Unity, and the "Weight of each Cubic Foot of its

Material and that of its Loading, Dnitj GDI

Tahle IV. — Mechanical Properties of the Materials of Construction . . li^i
Table V.— Uaefiil Nombera 698

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, Google

MECHANICAL PRINCIPLES

CIYIL ETi&OEEEING.

F^RT I.

STATIOS.

1. FoECE is that whieli tends to cause or to destroy
motion, or which actually causes or destroys it.

The difeopmi of a force is that straight line in which it
tends to cause motion in the point to which it is applied, or
in wliicli it tends to destroy the motion in it.*

"When more forces than one are applied to a body, and
their respective tendencies to communicate motion to it
counteract one another, so that the hody remains at rest,
these forces are said to be in equilibrium, and are called

PBESSUEES.

It is found by experiment \ that the effect of a pressure,
when applied to a solid body, is the same at whatever point
in the line of its direction it is applied ; so that the condi-
tions of the equilibrium of that pr^sure, in respect to other
pressures apphed to the same body, are not altered, if, with
out altering the direction of the pressure, we remove its
point of application, provided only the point to which we
remove it be in the straight line in the direction of which it
acts.

The science of Statics ie that which treats of the eqwli-
hrium of pressures. When two pressures only are a

■> Note (a) Ell. Appendix. ] Note (6) Ea. Appendix,

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2 THE fSIT OF PEESSCEE.

a body, and liold it at rest, it is found "by expei'iment tbat
these pressures act in opposite directions, and have their
directions always in the same straight line. Two such pres-
euroa are said to be equal.

If, instead of applying two pressui'es which are thus equal
in opposite directions, we apply them both in the same
direction, the single pressure which must be applied in a
direction opposite to the ^100 to sustain them, is said to be
double of either of them. If we take a third pressure, ecLual
to either of the two first, and apply the three in the same
direction, the single preeeure, which mnst he applied in a
direction opposite to the three to sustain them, is said to be
tr^>le of either of them : and so of any number of pressures.
Thus, fixing upon any one pressure, and ascertainiag how
many pressures equal to this are necessary, when applied in
art opposite direction, to sustain any other greater pr^sure,
we ai-rive at a true conception of the amount of that greater
pressure in terms of the first.

That single pressure, in terms of which the amount of any
other greater pressure is thus ascertained, is called an i'nit
of pressm-e.

Pressures, the amount of which are determined in terms
of some known unit of pressure, are said to be memured.

Different pressures, the amounts of which can be deter-
mined in terms of the smw imit, are said to be com/menmir-
ahle.

The units of pressure which it is found most convenient to
use, are the weights of certain portions of matter, or the
pressures with which they tend towards the centre of the
earth. The units of pressure are different in difl'erent coun-
tries. With UB, the unit of pressure from which all the rest
are derived is the weight of 22'815 * cubic inches of distilled
water. This weight is one pound troy ; being divided into
5760 equal parts, the weight of each is a grain troy, and
'7000 such grains constitute the pound avoirdupois.

If straight lines be taken in the directions of any number
of pressures, and have their lengths proportional to the
numbers of units in those pressures respectively, then these
lines having to one another the same proportion in length
that the pressures have in magnitude, and being moreover
drawn in the directions in which those pressures respectively
act, are said to represent them in magmttide and direction.

* This etondard iviis fixed by Act of Parliament, in 1834. The temperature
of the water is supposed to be 62° Fahrenheit, tie Height to be taken in air,
and the barometer to stand at 30 inches.

, Google

THE PAEAXLELOGEAJI OF PEESSUBES. S

A system of pressures "being in equilibrium, let any num-
ber of them be imagined to be taken away and replaced by
a single pressure, and let thia single pressure be sucii that
the equilibrium which before existed may remain, then this
single pressure, producing the same effect in respect to the
oqailibrium that the pressures which it replaces produced, ie
said to be the eesultant.

The pressures which it replaces are said to be the compo-
nents of this single pressm'e ; and the act of replacing them
by such a single pressure, is called the composition of

If, a single pressure being removed from a system in equi-
librium, it be replaced by any number of other pressures,
such, that whatever effect was produced by that which tbey
replace singly, the same effect {in respect to the eonditione of
the equilibrium) may be produced by those pressures con-
jointly, then is that single pressm-e said to have been eb-
SOLVED into these, and the act of making this substitution
of two or more pressures for one, is called the kbsolution
of pressures.

The Paballelogram of Pressuhes.

2, 1^ resultant of any two pressv/res allied to a point,
■is represented in direatlon J>y the diagon(A of a pa^<A-
lelogram, whose adjacent sidei represent those p>'easv/res m
magnitude and direction*

(Duchayla's Method. f)

To the demonstration of this proposition, after the excel-
lent method of Duchayla, it is necessary in the first place
to show, that if there be any two pressures P, and P, whose
directions are in the same straight line, and a third pressure
P, in any other direction, and il' the proposition be true in
respect to P, and P,, and also in respect to P, and P„ then
it wUl be true in respect to P, and Pj+P,.
Let Pj, P„ and Pj, form part of any system of pressui-es in
^ , „ equilibrium, and let them be applied to the point
'*':>.>,',"',C\ -A.; take AB and AC to represent, in magnitude
'^ ""*"-'*'^4 snd direction, the pressm^es P, and P„ and CD
a ^.^.^..9\ ^^ pressure P,, and complete the pandlelograms
OB and DP. Suppose the proposition ia be true with regard

* This proportion constitutes the founaation of the entire science of Statics.
\ Note (o) Ed. App.

./Google

to P, and P„ tlie reeultant of P, and P^ will then be in tlie
direction of the diagonal AT of the parallelogram BO, whose
adj acent sides AO and AB represent P, and P, in magnitude
and direction. Let P, and P, be replaced by this resultant.
It matters not to the eqinlibrium where in the line AF it is
applied ; let it then be applied at F. But thus appHed at
Fit may, without affecting the conditions of the eqnihbrium,
be in its turn replaced by (or resolved into) two other pressures
acting in CF and BF, and these will manifestly be eqnal to
Pi and Pj, of which P, may be transfeiTed without altering
the conditions to 0, and Pj to E. Let this be done, and let
P, he transfeiTed from A to C, we shall then have P, and
Pj acting in the dii-ections CF and CD at 0 and P„ in the
du-ection FE at E, and the conditions of the equiHbrinm will
not have been anected by the transfer of them to these
points. Now suppose that the proposition is also true in
respect to P, and P, as well as Pi and Pj. Then since OF
and CD represent P, and P, in magnitude and direction,
therefore their resultant ie in the direction of the diagonal
CE. Let them be replaced by this resultant, and let it be
transferred to E, and let it then be resolved into two other
pressures acting in the directions DE and FE; these will
evidently be P, and P^. We have now then transferred aU
the three pressures P^, P„ P,, from A to E, and they act at E
in directions parallel to the directions in which they acted at
A, and this has been done without affecting the conditions of
the equilibrium ; or, in other words, it has been shown that
the pressures P„ P„ P„ produce the same effect as it re-
spects the conditions of the equilibrium, whether they be
apphed at A or E, The resulta/n,t of P^, P^, P,, must there-
fore produce the same effect as it regards the conditions of
the equilibrium, whether it be applied at A or E. But in
order that this resultant may thus produce the same effect
when acting at A or E, it must act in the straight line AE,
because a pre^ure produces the same effect when applied at
two different points only when both those points ai'e in the
line of its direction. On the supposition made, therefore,
the resultant of Pi, P„ and P„ or of P, and P, + P,
acts in the direction of the diagonal AE of the parallel-
ogram BD, whose adjacent sides AD and AB represent
P, + Pj and P, in magnitude and direction ; and it has been
shown, that if the proposition be true in respect to Pi and
Pj, and also in respect to P, and P„ then it ]& t^e in respect
to Pi and Pj + P,. Now this being the case for all values
of P„ P„ P3, it is the case when Pi, P„ and P,, are equal

./Google

OF PEESSUEE8.

to one anotlier. But if P, be eq_ual to P, their resultant
■will manifestly have ite direction as much towards one of
these pressTU'es as the other ; that is, it will have its direc-
tion-midway between them, and it will bisect the angle BAG :
but the diagonal AF in this case also bisects the angle BAC,
since P, being equal to Pj, AC is ec^ual to AB ; so that in
this particulai- case the dii-ection of the resultant is the,
directiou of the diagonal, and the proposition is ti-ne, and
shnilarly it is ti'ue of P, and Pj, since these pressm'es are
equal. Since then it is true of P, and P, when they are
equal, and also of P^ and P„ therefore it is true in this case
of P, and P, + P„ that is of P, and 3 P,. And since it is
true of P, and P,, and also of P, and 2 P„ therefore it is ti-ae
of P, and P, + 2 P„ that is of P, and 3 P. ; and so of P, and
m, P„ if m be any whole number ; and similarly since it is
true of m P, and P,, therefore it is true of mP^and 2P„&c,,
and of otP, and «.Pj where n is any whole number. There-
fore the proposition is true of any two pressm-es mPj and
n P, wliich are co^mnensu/rdhle.

It is moreover true when the pressures are iiXr-
y'''%^~f coin/menmJ/rahle. Por let AC and AB represent
^•^{i:::^^ any two such pressures P, and P, in magnitude
and direction, and complete the parallelogram
■ ABDO, then will the dii'ection of the resultant of F, . and
P be in AD ; for if not, let its direction be AE, and draw
E(J pai'allel to CD. , Divide AB into equal parts, each less
than GC, and set off on AC parts equal to those from A
towards 0. One of the divisions of these will manifestly
fail in GC. Let it be H, and complete the parallelogram
AHFB. Then the pressure Pj being conceived to bo
divided into as many equal units of pressure as there are
equal parts in the line AB, AH may be taken to represent a
pressure P, containing as many of these units of pressure
as there are equal pai-ts in AH, and these pressures P, and
P, will be comfnen.swraUe, being measured in terms of the
same unit. ■ Their resultant is therefore in the direction AP,
and this resultant of P, and P, has its direction nearer to
AG than the resultant AE of P, and P, has ; which is
absurd, since P, is gi-eater than P,.

Therefore AE is not in the direction of the resultant of
P and P, ; and in the same manner it may be shown that no
other than AD is in that direction. Therefore, &c.

./Google

THE PKINCIPLES OF THE

3. The T'istdtant of two pressures applied in any direotiom
to a point, is represented vn magmtude as well as in direc-
tion oy the diagonal of the paraUelogram whose adjacent
sides represent those pressures in magnitude arid- m di/rec-
tion.

Let BA and CA repi-esent, in magnitude and
■-■•-. direction, any two pressiir^ applied to the point

f A. Complete the parallelogram BC. Then hy
0 the last proposition AD will represent the result-
* ant of these pre^uree in direction. It will also
t it in magnitude ; for, produce DA to d; and con-
ceive a pressure to be applied in G-A equal to the r^ultant
of BA and CA, and opposite to it, and let thie preesure be
represented in magnitude by the line GA. Then will the
pressures represented by the lines BA, CA, and 6A, mani-
iestly be presaui'es in equiHbnura. Complete the parallelo-
gram BG, then is the resultant of GA and BA in the
direction FA; also since GA and BA are in equilibrium
with OA, therefore this resultant is in equilibrium with CA,
but when two pressures are in equilibrium, their du-ections
are in the same straight hue j^ therefore FAC is a straight
Une. But AC is paraflel to BB, therefore PA is pai'allef to
BD, and FB is, by construction, parallel to GD, therefore
AFBD is a paraUelogram, and AD is equal' to FB and
therefore to AG. Bnt AG represents the resultant of CA
and BA in magnitude, AD therefore represents it in mag'ni-
tude. Therefore, &c.*

TuE PEmcirLE of the Equality of jVIoments.

4. DEFinrnoN. K any number of pressures act in the
same plane, and any point be taken in that plane, and per-
pendiculars be drawn from it upon the directions of all these
pressures, produced if neceseaiy, and if the number of units
m each pressure be then multiplied by the number of units
in the corresponding peii>endicular, then this product is
called the mommi, of that pressm'e (Aout the point from
which the perpendiculara are drawn, and these moments ai'e
said to be measured from that point.

* Sotc {(^) Eli. App.

./Google

5. If three pressures he in eqtdliirmm, and their inorihen.U
he taken about any point tn the plane in which they act,
then- the sum of the moments of those two pressures whioh
tmd to turn the plane in one direoHon about the point
from which the Tiuymmts are measii/red, is equal to the
mam&nt of tliat pressure which tends to turn it in the
opposite direction,

i'--^^. " 0 Let P„ P.„ Pj, acting in the directions
^^i7"l?\ I'.O, P,0, P,0, be any thi-ee pressures in

%...-.':^^^ ec[uilibriuni. Take any point A m the plane
*-"' in which they act, and measure their moments

from A, then will the sum of the momenta of Pj and P„
which tend to turn the plane in one direction about A, equal
the moment of P^ which tends to turn it iu the opposite
direction.

Through A draw DAB parallel to 0P„ and produce OP^
. to meet it in D. Take OH to represent P„ and take DB
such a lengtli that OD may have the same proportion to
DB that P, has to P,. Complete the pai-allelogram ODBC,
then wJU OD and 00 represent P, and P, in magnitude aud
direction. Therefore OB will represent P, in magnitude
and direction.

Draw AM, AW, AL, perpendiculars on 00, OD, OB,
and join AO, AC. Now the triangle OBO is equal to the
triangle OAC, since tliese triangles are upon the same base
and between the same parallels.

Also, A ODA+AOAB ^ AOBD = AOBC, ,.^ ,

^AODA+AOAB=AOA0, '^f^^r-Tt

.-. iODxAN+iOBxAL^^OCxAM, V^^l.
. ■ . P, X AN +^x AL=F, X AM. '""^

Now P, X AM, P, X AN, P, x AL, are the moments of P„
P„ P„ about A (Alt. 4,)

.■.m'P, + m'P,-m^P, (1).

Therefore, &c. &c.

6. K E be the resultant of P, and P„ then since E is
equal to P, and acts in the same straight line, m'E = mtP„
.•.m'P,+m'P, = m'E.

The sum of the moments therefore, about any point, of
two pressures, P, and P^ in the same plane, which tend to

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8 THE PEIXCIPLE OF THE

turn it in the same direction about that point, is equal tc
the moment of theii- Tesultant about that point.

K they had tended to torn it in oppcffiite dii-ections, then
the 3iffes'eno6 of their moments would have equalled the
moment of their resultant. For let E be the resultant of
Pi and P,, which tend to turn the plane in opposite direc-
tions about A, &c. Then is E equal to P^, and in the same
etraight line with it, therefore moment E ie equal to
moment P^, But by equation (1) m'P, — m*P, = m'P, ;
.■.m'P,— m'P, = m'R.

Generally, therefore, m' P, + m* P, = m* E (3),

the moment, therefore, of the restdtimt of arm two pressures
m the same plane is egual to the sum- or mfference of the
moments of its oomponents, acoordina as they aot to turn the
plane in the same direction aboiit the point from whwh the
mmrvents a/re measm-ed^ or in opposite directions.*

7. -27 any n/umber of pressures in the same plane ie in emd-
Ulyrium. a/nd a/rmf point J>e taken, i/n that plane, from
whAoh their moments are measv/red, th&n the sum, of the
moments of those pressures whdoh tend to twm the plane
in one d/i/retMon about thaipoint is equal to the sum qf the
moments of those which tend to twm it in the opposite

Let Pj, Pii Pj P«be any number of pressures in

the same plane which are in equi-
,^1 ^, ^ , librium, and A any point in the

^^f t^ " ^•'"'^1^;^,.--'^'^ plane from whicli their moments
1 '^v S-jy^ are measured, then will the sum of

ta the moments of those pre^ures

which tend to turn the plane in one direction about A equal
the sum of the moments of those which tend to turn it in
the opposite direction.

Let E, be the resultant of Pj and P„

E, E, and P„

E, E, and P„

■fee &c.

E^i E„_a andP„.

Tiierefore, by the last proposition, it being understood
that the moments of those ot the pressures Pj, P„ which
tend to turn the plane to the left of A, are to be taken neg*
tively, we have

' Kot« (c) Ed. App.

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EQUALITY OF

m' 1-i, = m' P, + m*^ P,.
m^ E, = m' K, + m^ P„
m' R, = m' R, + m' P.,
&c. = &c. &c.

m' Rft_i = m' E^_2 + m' P^ .

Adding these equations together, and striking out tlie
tei'ms common to both sides, we liave

m' R«_i ^ m' P, + m' P, + m* P, + + m^ P„

, . . (3), -where Rn— i is the resultant of allthe pressures P^
P„ . . . . P«.

But these pressures are in equilibrium ; they have, there-
fore, no resultant.

.■.E„_i = 0 .-. m'K,^i - 0,
.-.m' P, + m» P, -f m' P„ + m' P«= 0 . . . . (4).

Now, in this equation the moments of those pressures which
tend to turn the system to the left hatfd are to be taken
negatively. Moreover, the sum of the negative terms must
equal the sum of the positive terms, othei-wise the whole
sum could not ecLual zero. It follows, therefore, that the
sum of the moments of those pre^ures which tend to turn
the system to the right must equal the sum of the moments
of those which tend to turn it to the left. Tlierefore, &c. &c.

8. If any nwmher ofpresaw&i ousting in ths sameplime be m
eguUioru/m,, {mains')/ he vmaginm to be morndparaUd to
thevr eudstmg directit/ns, and all allied to the satnepomt,
80 as all to act v^pon thai point -m di/rections pa/ralld to
those in which they before acted upon different points, then
wUl they be in eqtdlibriiim about that point.

For (see the preceding figure) the pressure R, at whatever
point in its direction it be conceived to be applied, may be
resolved at that point into two pressures parallel and equal
to P, and P, : similarly, R, may be resolved, at any point in
its du'ection, into two pressures parallel and equal to E, and
Pj, of which R, may be resolved into two, parallel and equal
to P, and Pj, so that R, may be resolved at any point of its
direction into three pressures parallel and equal to Pi, P„ P^ :
and, in lite manner, R^ may be resolved into two pressures
parallel and equal to R, and P„ and therefore into four pres-
sures pai'allel and equal to P„ P^, P,, P„ and so of the rest.

./Google

ifJ THE POr.YGON

Therefore K^-i may, at any point of its direction be resolved

into n pressures parallel and equal to P„ P„ Pj, P„ ;

if, therefore, n such preesures were applied to that point,
they would just he held in ecLuilibrium by a pressure equal
and opposite to Eu-i. But E«_i = 0; th^e n presBures
would, thei'efore, be in equilibrium with one another if
applied to this point.

Now it is evident, that if, being thus applied to tMsvoiat,
they would be in equilibrium, they would be in equihbrium
if similarly apphed to any other point. Therefore, &c.

The Poltgoh of Peessubes.

9. The conditions of the equiUbriwn of any number of pres-
sures c^Ued to apoimt.

Let 0P„ OP5, OP3, &c., represent in mag-
nitude and direction pressures P„ P„ &c.,
applied to the same point 0. Complete the
parallelogram OP, AP„ and draw its diago-
nal OA ; tlien will OA represent in magni-
* tude and direction the resultant of P, and

P,. Complete the paralleloffl-am OABP„ then will OB
i-epresent in magnitude and direction the resultant of OA
and P, ; but OA is the resultant of P, and P„ therefore OB
is the resultant of P„ P,, P, ; similarly, if the parallelogram
OBCPj be completed, its diagonal 00 represents the result-
ant of OB and P., that is, of P„ P„ P„ P„ and in like
manner OD, the diagonal of the parallelogram OCDP„
represents the resultant of P„ P„ P^, P„ P^.

Kow let it be observed, that AP, is equal and paraliel to
0P„ AB to 0P„ BO to 0P„ CD to 0P„ so that P,A, AB,
BC, OD, represent P„ P„ P,, P^, respectively in magnitude,
and are parallel to their directions. Moreover, OP, is in the
direction of P, and represents it in magnitude, so that the
sides 0P„ P,A, AB, BO, CD, of the polygon 0P„ ABCDO
represent the pressures P„ P^, P„ P„ P„ respectively in
magnitude, and are paraliel to their directions; whilst the
side OD, which completes that polygon, represents the
resultant of those pressures in magnitude and direction.

If, therefore, the pressures P„ P„ P„ P., P„ be in eqiiili-
brium, so that they have no resultant, then the side OD of
the polygon must vanish, and the point D coincide with O.
Thus, then, if any number of pressures be applied to a point

./Google

and lines be drawn parallel to the directions of those pros-
Bures, and repreBenting them in magnitude, so as to form
sides of a polygon (care being taken to draw each line from
the point where it nuites with the preceding, towmds the
dh-ectiou in which the corresponding pressure acts), then the
line thus dj-awn parallel to the last pi'eesure, and representing
it in magnitude, will pass through the point fl.'om which the
polygon commenced, and will just complete it if the pres-
sures be in eiiuilibrium ; and it thej be not in equilibn\im,
then this last line will not complete the polygon, and if a
line be drawn completing it, that line will represent the
resultant of all the pressarea in magnitude and direction.

ITiis principle is that of the polygon of peessuees ; it
i»htains in respect to pressures applied to the same point,
whether they be in the same plane or not,

10. If (my nwnber of pressures in the srnne plane he m egup-
Ulyri/wm^ amd eaoh he resdI/eeA vn directions paraUel to amy
two rectangular (nees^ then the srnn of aU those resolved

presswres, whose tendmoy is to oonrniumcate motion i..

di/rection along eUher axiSf is equal to the swm of those
whose terhdenoy is m the o^^posite direction.

Let the polygon of pressures be formed in respect to any
number of pressm'eis, P„ P„ P„ P„ in the same plane and in
equilibrium (Arte. 8, 9), and let the sides of
this polygon be prmeoted on any straight line
Ate m the same plane. Now it is eyident,
that the sum of the projections of those sides
I of the polygon which ioi-m that side of the
figure which is nearest to A^, is equal to the sum of the pro-
jections of those sides which form the opposite side of the
polygon : moreover, that the fonner are those sides of the
polygon which represent pressures tending to communicate
motion from A towards x, or from left to right in respect to
the line Aib,' and the latter, those which tend to comm\mi-
cate motion in the opposite direction. Now each projection
is equal to the correspondir^ side of the polygon, multiplied
by the cosine of its inclination to Aic.> The sum of all those
sides of the polygon which represent pressures tending to
communicate motion from A towarde tc, multiplied each by
f he cosine of its inclination to Aic, is equal, therefore, to the
sum of all the sides representing pressures "whose tendency
is in the opposite direction, each being similarly multiplied
by tlie cosine of its inclination to Ax. Now the sides of the

./Google

12 THE EEeOLUTION

polygon represent the preesiires in magnitude, and are
inclined at the same angles to Ai8, Therefore, each preesore
being miiltiplied by the cosine of its inclination to Aic, the
sum of all these products, in respect to those ■which tend to
communieate motion in one direction, equals the sum simi-
larly taken in respect to those which tend to communicate
motion in the opposite direction ; or, if in taking this sum it
he nndeiBtood that each term into which there enters a pres-
sure, ■whose tendency is from A towards a), is to he tahen
poeitively, whilst each into which there enters a pressure
which tends from <s towards A is to be taken negatively,
then the sum of all these terms will equal zero j that is,
""'^■"■j the inclinations of the directions ot P„ Pj, Fj . . . P„

. a„ respectively,

P, COS. a. + P^COS.a, + P,C03.O, + ....+ P« cos.<v =0 . . . (5),

in which expression ail those terms are to be taken negar
lively which include pressures, whose tendency is from x

This proposition being true in respect to any axis, Aa) is
true in respect to another axis, to which the inclinations of
the directions of the pressures are represented by 0„ j3„ /i„
/3a , so that,

P. cos. 0, + P, cos. /?,+ ....+ P„coa. P„=0.

Let this second axis be at right angles to the first :

= ein. a„ &c. = &c.

.-. Pj ein. a, + P, sin. a^ + + P« sin. «™ = 0 (6) ;

those terms in this equation, involving pressures which tend
to communicate motion in one direction, in respect to the
axis Ay being taken with the positive sign, and those which
tend in the opposite direction with the negative sign.

If the pressures P„ P,, &c. be each of them resolved
into two others, one of which is parallel to the axis Ax, and
the other to the axis A^, it is evident that the pressurea
thus resolved parallel to Ase, will be represented by F, cos. o„
P, cos. «j, &e., and those resolved pao-allel to Ay, by
, P, sin. a^ P, sin. a„ &c. Thus then it follows, that if
any system of pressures in equilibrium be thus resolved
pai-allel to two rectangular axes, the sum of those resolved
pressures, whose tendency is in one direction along either

./Google

13

axis, is equal to the sum of those whose tendencj is in the
oppctsite direction,*

This condition, and that of the equality of momenta, are
necessary to the equilihrium of any number of pressures ic
the same plane, and they are together sujiment to that equi-
librium.

11. To determine the remltawt of any nuinier (>f ^essures
in the same ^Itme.
.1 If the pressures P, P, . . . . P„he not in

^■—^^ equilihrium, and have a resultant, then one

' '' side is wanting to complete the polygon of

pressure, and that side represents the res-
ultant of all the pressures in magnitude.

JCfli

'■ LLLL£liilli Ul ail LUC ^^ICOOUICD LU limgLl-l u u u.i:7,

and is parallel to its direction (Art. 9).
Moreover it is evident, that in this case the sum of the pro-
jections on Aic (Art, 10) of those lines which form one
side of the polygon, will be deficient of the sum of those of
the lines which form the other side of the polygon, by the
projection of this last deficient side ; and therelore, that the
sum of the resolved pre^ures acting in one direction along
the line Asm, will be less than the sum of the resolved pres-
sures in tiie opposite direction, by the resolved part of the
resultant along this Hne. N^ow if E represent this resultant,
and Q ite inclination to Aro, then E cos, & is the resolved part
of E in the direction of Aas. Therefore the signs of the terms
being understood as before, we have

E COS. e=P, cos. a, + Pj COS. o,-(- .... +P„cos. «» . . (t).
And reasoning similarly in respect to the axis Ay, we have
R sin. fl=P, sin. «.+P, sin. o,-l- . . . . -I-P^sin. a^ . . . (8).

Squaring these equations and adding them, and observing
that E° sin.' ^-fE'cos." 6=E' (sin.'ff-fcos.'ff) = E°, we have

E'=(SP sin. oy + (,Sp COS. «y (9),

■where 2P sin. a is taken to represent the sum P, sin. a, -r
Pj sin, dj-l-P, sin. a,-)-&c,, and sp cos. a to represent the
sum P, cos. ffl,+P, COS. «5 + P, COS. a^-V &c.

Dividing equation (8) by equation (7),

tan.e=^^ (10).

SP COS. o '

Thus then by equation (9) the magnitiide of the resultant
" Xotc (/) Ed. App.

, Google

14 THK PAEALLELOrLPEDOH

E is known, and- by ec[nation (10) its inclination 9 to the axia
A* is known. In order completely to determine it, yiG have
yet to find the perpendicular distance at which it acts from
the given point A. Por this we must have reconree to the
condition of the ec^uality of moments (Art. Y).

If the sum of the moments of those of the pressures, P^
P P„ , which tend to turn the system m one direc-
tion about A, do not equal the sum of the moments of those
which tend to turn it me other way, then a pressure heing
applied to the system, equal and opposite to the resultant K,
will bring about the ecLuality of these two sums, so that the
moment of E must be equal to the difference of these sums.
Let then p equal the pei-pendicular distance of the direction
of E from A. Therefore

E2i=m'P,+m'P,+m'P,+ .... +m'P„. . . (11),

in the second member of which equation the moments of

those pressui'es are to be taken negatively, which tend to

communicate naotion round A towards the left.

Dividing both sides by E we have

^^m.F. + m.F.+ ....+m^ ^^,^^^

Thna then by equations (9), (10), (12), the magnitude of
the resultant E, its inclination to the given axis Ai«, and the
perpendicular distance of its direction from the point A, are
known ; and thus the resultant pressure is completely deter-
mined in magnitude and direction.

The PAEAiLELOriPEDOS OF PBKSSrEES,

13. Three pressv/res, P^, P„ P^, hmig mmUed to the same

pomi A, in directions xA, yA, sA, whtch are not in the

aamieplcme, it ia re^/uired to detemvme their resvUant.

Take the lines P, A, P, A, P, A, to represent the pressures

P„ P„ P„ in magmtude and direction.

Complete the parallelopipedon EPjP.P,,

ofwbichAP„AP5,APs, are adjacent edges,

and draw its diagonal EA ; then will RA

' represent the resultant of P„ P,, P,, iu

direction and magnitude. For since

P,SP,A is a parallelogram, whose adjacent

sides Pi A, P, A, represent the presiires

P, and P, in magnitude and direction, therefore its diagonal

./Google

OF THKEli PEESSUEES, 15

8A represents the reanltant of these two pressures. And
similarly RA, the diagonal of the parallelogram ESAPj, re-
pr^enta in magnitude and direction the resultant of SA and
P„ that is, of i*„ P, and P^, since SA is the resultant of
P, and P,.

It is evident that the fourth pressure neceasaiyto produce
an equilibrium with P„ P„ P,, heing equal and opposite to
their resultant, is represented in magnitude and direction
byAE.

13. Three pressv/res, Pi, P., P„ lemg m eqmUbrivm., it is,
required to determine the third F, in terms of the other
ttM, a/nd their inclination to one anoth&r.

Let AP, and AP, represent, the preesui-^ Pj and P, in
magnitude and aireetion, and let the inclination
! ...,^^ ^^ P, AP, of P, to P, be represented by ,fl,. Oom-
■■fC i plete the pai'allelogram AP, itP„ and draw its
1^,^ diagonal Alt. Then does Alt represent the
' resultant of P, and P, in magnitude and direc-

tion. But this resultant is in equilibrium with P^, since P,
and P, are in. eqaillbrium with P,. It acts, therefore, in the
same straight line with P„ hut in an opposite direction, and
IB equal to it. Since then AR represents this resultant in
magnitude and direction, therefore RA represents Pjin mag-
nitude and direction,

Now, A^=AP?— 2Ap; . RE . cos. AP.E-f^';
also, AP.K=7r— P.AP.^TT-— ,9„ P,E=AP'„ and AP„ AP„
AR, represent P„ Pj, Pj, in magnitude.

.-. P,'=P,''— 2P,P, cos. (t— .flO+P/-

Now cos, (^— ,e,)= —COS. ,e„ ;. P,'=P,'+3P,P,cos. A+^A

.-. P,= VP,' + 2P,P, COS. >e,+P,' (13).

14, If three presswres, P„ Pj, P„ he in eqidlibrium, a/ny two
of them are to one another in/versely as the sines of their
indmaiions to the third.

Let the inclination of P, to P, be represented by .S,, and
that of P, to P, by A-

Now P,AR=Tr— P,AP,=TT— fl„ .-. sin. P.AR^sin, fi,;
P,RA=P,AR='^— i'AP^^^—A, " sin. P,RA=sin. ,6,.

./Google

16

OF PAEALLFL PEEBBUBEB.

Also,

AP, AP, sin. P,KA
AP, P.E - sin. P,AK'

... P._it-^ ,

P, ~ Bin. ,9,

(14).

That is, P, is to P, inversely, as the sine of the inclina-
tion of Pi to Pj is to the sine of the inclination of P, to Pj.
Therefore, &c. &c. [q. e.b.]

Of Paeallel Pebssuees.

15. The principle of the equality of moments obtains in
respect to pressures in the same pla/ne whatever m,ay ie
their molvnaiions to one cmothm\ emd therefore if their
inolma^ons ie mfimtely small, or if they l>epatrallel.

In this case of parallel pressures, the same line AB, which
^ ie drawn from a given point A, pei'pendicular
to one of these pressnres, is also perpendienlar
to ^ th6 rest, 6o that the perpendicnlai^ are
T?^\ here the parte of this line AM,, AM,, &c.
^ ' intercepted between the point A and the direc-
tions of the pressuree respectively. The principle is not how-
ever in this case true only in respect to the intercepted parts
of this perpendiculai- line AB, hnt in respect to the inter-
cepted parts of any line AO, drawn through the point A
across the directions of the pressnree, since the intercepted
parts Awi,, A.m„ Am.,j &c. of mis second hne ai'e proportional
to those, AMj, AM,, &e, of the first.

Thus taking the case represented in the figure, since by
the principle of the eq^iiality of moments we have,

AM, . P, + AM. . F,=AM, . P.+AJS,P,+AM,P, ;
dividing both sides by AM^,

AM, AM^ _Ark T> J^'

AM, ■^■+ AM, -^'-AM^ -^'^ am; • ^= + ^0
AM, Am, AM, Am,
Bntbysimilartriangles, ^^^=J^; am:=A^; &«■=&<>■

. ^ p+^ p^Ato, p , Am, p
"■ A-nij ■ ' Am^ ' ' Am, ' ° Am^ ' " '*
Therefore multiplying by Am,,

A^ . P,+ Am; . P.=A^ . P,+Am; . P,+A^. P,.
Therefore, &c. [q.e,d.]

./Google

.KL PEBaSrEES.

16. ToJindtheT&siiltantfffanymiml>&r of paraUel pressures
m the samis flwne.

It i8 evident tliat if a pressure eqnal and opposite to the
resultant were added to the eyatem, the whole would be in
equilibrium. And being in equilibrium it has been ahown
(Art. 8.), that if the pressures were aU moved from their
present points of application, so as to remain parallel to their
existing directions, and applied to the same point, they are
such 83 would be in equilibrium about that point. But
being thus moved, these parallel prassures would all have
their dii'eetions in the same straight line. Acting therefore all
in the same sti-aight line, and being in equilibnum, the sum
of those pressures whose tendency is in one direction along
that line must equal the sum of tliose whose tendency is in,
the opposite difection. Now one of these sums inchides the
resultant li. It is evident then that before K was introduced
the two sums must have been unequal, and that R equals the
excess of the greater sum over the less ; and generally that if
SP represent the sum of any number of pai'allel pressures,
those whose tendency is in one direction being taken with
the positive sign, and those whose tendency is in the opposite
direction, with the negative sign ; then

E = SP (15).

the sign of E indicating whether it act in the direction of
those pressures which are taken positively, or those which are
taken negatively.

Moreover since these pressures, including R, are in equi-
librium, therefore the sura of the mommis ahoYit any pomt,
of those whose tendency is to communicate motion in one
direction, must equal the sum of the moments of the rest — ■
these moments being measured on any line, as AO ; but one
•5 ^^ ^ ofthesesumsincludesmemoraentofE; these
^vi-wT ' ''''^^ sums must therefore, before theintroduc-
A-c'&^Q.V.. tion of E? liave been unequal, and the moment
V^'^A" of E must be equal to the excess of the greater
^ ^ sum over the less, so that, representing the
sum of the moments of the pressures (E not being included)
by s m' P, those whose tendency is to communicate rootiou
in one direction, having the positive sign, and the rest the
negative ; and representing by a> the distance from A, mea-
sured along the line AC, at which E intersects that line, we
have, since xK is the moment of E, icE = s m' P, where the

./Google

' PAEALLEL PaESSUEES.

sign of wB. indicates the direction in which E tends to tni
the system about A, "but E = 2;P,

2P

. (16).

Equations (15) and (16) determine completely the magai-
tude and the direction of the resultant of a system of parallel
pressiu'es in the same plane.

IT. To determme the restdtani of any number of jpa/rallel
pressures not m the same plane.

Let P, and P, he the points of application of any two of
..^^ these pressures, and let the pre^nree themselves
f}6^ he represented by P, and P^. Also let their
^i^f"^ ... resultant K, intersectthelinejoining the points
/iJ^'"^ P, and P, in the point R, ; produce the line
'■■'"* P„ P,, to intersect any plane given in position,

in the point L. Through the points P„ P,, and K„ dravf
P,M„ P,M„ and K,N", perpendicularly to this plane : these
lines will be in the same plane with one another and witJi
PjL : let the intersection of this last mentioned plane with
the first be LM„ then wffl P,1VI„ P,M„ and R.K, be per-
pendiculars to LMj ; moreover by the last proposition,

P,LP, + P,LF, = E,I:R.;

. ^ LP LP,_

But by similar triangles

LP,_P^,
LIt~E,K=
. V S'-^P £^

LP._PA
LR~R,N,

Let now the resultant, R,, of E^ and P,
intersect the line joining the points R, and
P, in the point R„ and similarly let the
resultant, R,, of E, and P, intersect the
ine joining the points R, and P^ in the
point R„ and so on : tlien by the last equa^
tion.

./Google

r PAR4I.I.EL PKiiSSUEES.

P, . PjM,+P, . I\M, = E, KJJ, .
Similarlj, K, . B,«, +F, . F,«. = B, B,M,.
B, . ETN, + P, . P,M, = E, EX,
&c. + &c. = &c.
E,-, . E,_,N^,+P, .paE.=E.--, . E»-, iN»
Adding these eq^uationSj and striking out tenns
both sides,
p, . p;m;+p,Tvm,+ . . . +P..P3.=B^, ."SZ

-All)

Now,

B,=P, + P., B,=E,+P.=P,+P,+P„
E,=E,+P,=P,+P, + P, + P., &c.=&o.

E,^,=P, + P, + P,+ +P.;__

. P,M,+P, .

.'. E^i iV, . F, + P.+F. + &c. + F.;
P,M.+ +P. . P»M.;

.•.B_,H».

F. p,m:.+p.p.m.+

. +P. . FJt

P.+F,+P,+ . . . +F.

(18);

in whicli expression those of the parallel pressures P„ Pj,
&c. which tend in one direction, are to be taken positively,
whilst those which tend in the opposite direction are to be
taken negatively.

The line 'Rn-i ^n-i represents the perpendicular distance
from the given plane of a point through which the resultant
of all the pressures P„ P, . . . . P„, passes. In the same
manner may be detennined the distance of this point from
any other plane. Let this distance be thus determined in
respect to three given planes at right angles to one another.
Its actual position in space will then be known. Thus then
we shall know a point tlirongh which the resultant of all the
pressures passes, also the direction of that resultant, for it is
parallel to the common direction of all the pressures, and we
shall know its amount, for it is equal to the sum of all the
pressures with their proper signs. Thus then the resultant
pressure will be completely known. The point Rn^i is called
the Centke of Pakallei, PnEasTTRiss.

18, The product of any pressure by its perpendicular dis-
tance from a plane (or rather the product of the number of
units in the pressure by the number of units in the perpen-
dicular), is called the moment of the presswe^ in, respect to
that plane. Whence it follows fl'om equation (IT) tnat tJm
sum of the moments of <my numier of paraUd pressures m

./Google

respect to a given plame ia equal to the -nior/ieiit of their
reeuUami in respect to that plane.

19. It is evident, from equation (18), that tlic distance
Kn_i Nn_i of the centre of pressiifre of any number of
parallel preseures from a given plane, m independent of the
directions of these pardlel pressures, and is dependent
wholly upon their amounte and the perpendicular distances
P,M,, P^3, &c. of their points of application from the
given plane.

So that if the directions of the pressureB were changed,
provided diat their amounts and points of application
remained the same, tkew centre of p^resswe, determined as
above, would remain unchanged; that is, the resultant,
although it would alter its direction with die directions of
the component pressures, would, nevertheless, always pass
thi'ough the same point.

The weights of any number of different bodies or diiFerent
parts of the same body, constitute a system of parallel pres-
sures ; the direction, therefore, through this system of the
resultant weight may be determined by the preceding pro-
position ; their centi'e of pressure is their centre of gramty.

The Centke of Geavity.

20. The remdtant of the weights of amy n/umher of bodies
or mm-ts of the same l/ody wnited into a system, of w/oor
riable form passes through the sams -point in it, into what-
ever position it may be twned.

For the effect of turning it into different positions is to
cause the directions of the weights of its parts to li'averse
the heavy body or system in dirorent directions, at one time
lengthwise for instance, at another across, at another
dbhquely ; and the effect upon the direction of the resultant
weight through the body, produced by thus turning it into
diffei-ent positions, and thereby changmg the directions in
which the weights of its component parts traverse its mass,
is manifestly the same as would be proaueed, if without alter-
ing the ptffiition of the body, the direction of gramity could
be chcmged so as, for instance, to mal^o it at one time tra-
vei-se ttiat body longitudinally, at another obliquely, at a
third ti-ansversely. But hy Article 1&, this last mentioned
' ' ' 3 the common direction of the parallel pres-

,y Google

THE CESTKE OF ©KAVnT. 31

STires through the "body without altering their amounts or
their points of application, would not alter the position of
their centre of pressure in the hody; therefore, neither would
the first mentioned change, whence it follows that the
cen^e of pressure of the weights of the parts of a heavy
hody, or of a system of invanable foiin, does not alter its
pteition in the "body, whatever may be the position into
which the body is tuiTied; or in other words, that tlie
resultant of the weights of its pai-ts passes always through
the same point in the body or system ia whatever position
it may be placed.

Tine point, through which the resultant of the weight* of
the parte of a body, or system of bodies of invariable fonn,
passes, in whatever position it is placed ; or, if it he a body
or system of vaa-iiMe form, through whidi the resultant
wotitd pass, m whatever position it were placed, if it became
rigid or invaiiable in its form, is called the CsNTiiB oe

(tEA-VITY.

21. Since the weights of the parts of a body act in
parallel directions, and all tend in the same direction, there-
fore their resultant ia equal to tiieir sum, Now, the result-
ant of the weights of the parts of the body would produce,
singly, the same effect as it regai-ds the conditions of the
equihbrinm of the body, that the weights of its parts
actually do collectively, and this weight is equal to tlie sum
of the weights of the paits, that is, to the whole weight of
tlie body, and in every position it acts vertically downwards
through the same point in the body, viz. the centime of
gravity. Thus then it follows, that m miery position of the
oody amd %mder every oirciimstamos, the weights of' its pwrts
frod/uce the same ^ect in respect to the condiiMons of its
equilibrium, as though they were aU collected in <md' acted
through that one point of it — its cewtre of gravity.*

rt importaat usea in the mechanism of the uniTerse, ai
e of tha arts; aoothev proof of it is therefore subjoined, which
re satisfoctovj to aome readers than that giTen in the text. The
ajstem being rigid, tlie distance Pi, Pj, of tlie points of
■I application of any two of the preeaures remains the

1 1 same, into whatever position the boil t may be turned ;

I the only difference produced in tlie circumstance under

^ which they are applied is an alteration in the indina-

\ tions of these pressures to the line Pi, P> ; now being

f \ weights, the dircelions of these pressures always remain
\v parallel to one another, whatever may be tlieir inclina-
tion ; thus then it follows by the principle of the equa-

, Google

OF CKATITr.

{2. To determine the posiUo.'^ of the centre of gramty of
two weights, P, andr^forni'mg^aTt of a rigid systmi.

Let it be represented by G. Then since the resultant of

^0 ^ Pi and P, passes througb G, we have by eq^oa-

' * ^ tion (16), taking Pj as the point fi-om which the
s are meaBui-cd,

■■ ' P, + P,
whence the position of G is known.

23. It is required to determine the centre of gravity of three
weights P„ P„ P,, not in the same straight Une, cmdf&rmr
ingpm't of a rigid system.

Find the centre of gravity G„ of P, and P^, as in tlie last

proposition. Suppose the weights P, and P, to

ri be collected in tf„ and find aa before the com-

^Jo, mon centre of gravity G^ of this weight P,+P„

j/r"^^] so collected in G„ and the third weight P^. It

L is evident that this point G, is the centre of

gravity required. Smce G, is the centre of

gi-avity of P, and P,+Pa collected in G„ we have by the

laBt proposition

g;g-, . p,+p,+^^g;p. . P„

. -^,_ G.P,.P,
.. '^''-^^-p^+p^+p;

lity of momenlB {Art. 15), that Pi+Pj .PiRi^Pa . PiPa, so that for every
Buoh molination of the pressures to Pi Pi, the Une PiEj is of the same length,
and the point E, therefore the same point; therefore, the Une PjRi is always
the same line io the body; and E, which equals Pi+I"!) is always the same
pressure, as also is Ps, and these prespurea always remtda parallel, therefore,
for the same reason aa before, Ki is always the same point in the body in

whatever position it may be turned, and so of B|, &t and R»-i. That

is, in every position of the body, the resultant of the ireiglits of ils parts
passes through the same point R>..i in it. Since the resultant of the weights
of the parts of a body always passes through its centre of grayity, it is
evidect, that a single force applied at that point equal and opposite to this
resultant, that is, equal in amount to the whole weight of the body, and in a
direction vertioally upwards, would in every position of the body sustain it.
This property of the centre of gravity, viz. that it is a point in the body whert
■ single force would support it is sometimes taitcn as the definition of it.

, Google

!■ A TRIANGLE.

If P„ P„ P„ be all equal, then
Moreover in this ease,

PA=iP7^

24. Tojind the centre of gravity of four weights not in thf
same^lane.

Let P„ P,,

find the
P» P..

, P„ represent these ■
centre of gravity G, of the t
P„ as in the last proposition ; enpp<ffie tnese
three weights to be collected in G„ and then
Und the centre of gravity G, of the weight
thus collected in Q-, and "p,. G, will be the
centre of gravity reqtiired, and since G, is
tlie centre of gi-avlty of P, acting at the
point P„ and of P, + P,+P, collected at G„

G,G,

.P, + P,+Pj-P,=G,P,.P„

P,+P,+P,+P/
aee weights be equal, then by the above equation,

also, " G A'^i G,P„
and G,P,=iP,P,.

25. 1

i CENTRE OF GRAVITY OF I

Let the sides AB and PC of the triangulai- la/mi/rm ABC

be bisected in E and D, and the lines CE and

\ AD drawn to the opposite angles, tlien is the

J..^ intersection G of these lines the centre of gravity

7^ \ of the triangle : for the triangle may be supposed

/ v\ *^ ^^ made up of exceeding/ narrow rectangular

n^ ^~^ strips or bands, parallel to 150, each of which will

be bisected by the line AD ; for by similar tiiangles

PR : DB :: AE : AD :: EQ : DO, tlierefore, altemando,

PE : EQ : : DB : DC ; but DB=DO ; ' tlierefore PE=EQ.

Therefore, each of the elementary bands, or rectangles
parallel to EC, ■which compose the triangle ABC, would
separately balance on the line AD ; therefore, all of them

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24 THE CENTEE OF

joined together would balance on the line AD, therefoi'e the
centre of gravity of the triangle is in AD.

In the same manner it may he shown that the centre of
gravity of the triangle is in the line CE ; therefore, the cen-
tre of gravity is at the intersection G of these lines.

Now DG=-t DA : for imagine the triangle to be without
weight, and tlu-ee eqiial weights to be placed at the angles
A, B, and 0, then it is evident that these tluee weights will
balance upon AD ; for AD being supported, the wei^t A
will be supported, since it ie in tnat line ; moreover, B and
C will be supported since they are equidistant from that
line.

Since, then, all three of the weights will balance upon
ADj their centi-e of gravity is in AD. in like manner it
may be shown that the centre of gravity of all three weights
is in CE ; therefore it ie in G, and coincides with the centre
of gravity of the triangle.

Now, suppose the weights B and 0 to be collected in then-
centre of gravity D, and suppose each weight to be repre-
sented in amount by A, a weight equal to 2A will then be
collected in D, and a weight equal to A at A, and the centre
of gi-avity of these is in G ; therefore DA x A = DG- X
(9A + A),

. • . D A = 3 DG, or DG = i DA.* [q.i=,d.]

. THE CENTRE OF GEAVrTY OF THE FTRAIOD.

Let ABC be a pyramid, and s^ippose it to be
made up of elementary laminse hca, pai'allel to
the base BCD. Take G, the centre of gravitv
of the base BOD, and join AG; thenAGwill
3s through the centre of gravity g of the
lamina lcd,j: thei-efore each of the laminte will separately
balance on the straight line AG ; therefore the laminae when
combined will balance upon this line ; therefore the whole
figure will balance on AG, and the centi-e of gravity of
the whole is in AG. In like manner if the centre of ^'avity
H of the face ABD be taken, and CH be joined, then it raaj
be shown that the centre of gravity of the whole is in CH ;

" Note {^) Ed. App.

f For produce tha plane ABG to intersect the plane ADC in AM, then by
similiir tritmgleB DM : MC : : >foi : mn, but DM — MC ; therefore dm — me. Also
by similar trinngles GM ; BM;;^^,: Sm, but QM — i BM; therefore ^^i
ftra. Since then (Jm—iifcandjni—iim, therefore 3 is the centre of gravltj
of the triangle bdc.

, Google

therefore the linea AG and CH intersect, and tiie eentro of
gj-avitj is at their intereection K.

ilfow GK is one-foiirth of Or A. ; for suppose eqiiiil weights
to be placed at the angles A, B, 0, and D of the pyramid
(die pyramid itself being imagined without weight), then
will these fonr weights balance npon the line AG-, for one
of them, A, is m that line, and the line passes thi'ough the
centre of gravity G of the other three.

Since, then, the equal weights A, B, 0, and J) balance
upon the line AG, their centime of gravity is in AG ; in the
same manner it may be shown that the centre of gravity of
the fom- weights is in CH, therefore it is in K, and coincides
with the centre of gi'avity of the pyi'amid.

Now let the number of units in each weiglit be repre-
sented by A, and let the three weights B, C, and D be
supposed to be collected in then- centre of gravity G ; the
four weights will then be reduced to two, viz, 3A at G, and
A at A, whose common centre of gravity is K,

.-. GKx3A+A = GAxA,

.-. 4GK = GA or GK = i GA.* [q.e,.d.]

27. The centre of gravity of a pyramid with a polygoiial hase
is situated at a vertioal hetght from, the hase, equal to one
fmurth the whole height of the pyramid.

Eor any such pyramid ABODEF may be supposecl to
be made up of triangular pyi'amids ABOF,
ACDF, and ADEF, whose centres of gravity
G, Hj and K, are situated in lines AL, AM,
and AN, di'awn to tlie centres of gravity L, M,
and N of their bases ; LG being one-fourth of
LA, MH one-fourth of MA, and NK one-fourth
of NA. The points G, H, and K, are therefore in a plane
parallel to the base of the pyramid, and whose vertical dis-
tance from the base equals one-fo\u'th the vertical height of
the pyramid.

Since then the centres of gravity G, H, and K of the ele-
mentary triangular pyramids which compose the whole poly-
gonal pyramia are in this plane, therefore the centre of gravity
of the whole ia in this plane, *. e. the centre of gravity of the
whole polygonal pyramid is situated at a vertical heimt from
tlie base, equal to one fourth the vertical height of the whole

* Note (;i) Ed. App,

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26 THE CENTRE OF GEAVTCY

pyramid, or at a vertical depth from the vertex, equal to three

iburthfi of the whole. Now the above proportion is true,

whatever be the number of the sides of the polygonal base,

and therefore if they be infinite in number ; and therefore it

true of the cone, which may be considered a pyramid hav-

g a polygonal base, of an infinite niimber of sides ; and it

trne whether the cone or pyramid be an (Mi^vs or a right

oone or pyramiid.

28. If a body be of a prismatic form, and symmetrical
about a certain plane, then its whole weight may be sup-
posed to be collected in the surface of that plane, and iini-
formly distributed tlu'ough it. For let
ACBEFD represent such a prismatic'
body, and d>c a plane about which it is
symmetrical : take m, an element of uni-

^ form thickness whose sides are parallel to

^ the sides of the prism, and which is
terminated by the faces ACB and DFE of the prism ;
it is evident that this element m -will be bisected by the
plane a5c, and that its centre of gravity will therefore
lie in that plane, so that its whole weight may be s^ip-
posed collected in that plane ; and this being true of
every other similar element, and all these elements be-
ing equal, it follows that the whole weight of the body
may be supposed to be collected in and uniformly dis-
ti-ibuted through that plane. It is in this sense only that we
can speak with accuracy of the weight and the centre of grar
vity of a plcme, whereas a plane being a surface only, and
having no thickness, can' have no weight, and therefore no
centre of gravity. In like manner when we speak of the
centre of gravitv of a curved surface, we mean the centre of
gravity ofa bod'y, the weights of all whose parts may be sup-
posed "to be collected and imiformly distributed throughout
that curved surface. It is evident that this condition is
approached to whenever the body being hollow, its material
is exceedingly thin. Its whole weight may then be conceived
to be collected in a surface equidistant from its two external
surfaces. Li like manner an exceedingly thin imiform cui^ved
rod may be imagined to have its weight collected unifonnly
in a line passing along the centre of its thickness, and in this
sense 'we may speak of the centre of gravity of a Une,
although a line having no breadth or thiclness can have nc
weight, and therefore no centre of gi'avity.

./Google

r QUADRILATEKAL FIGUKE.

29. THE CJi]^"TKG OF GEi^'lTY OF A TKAPEZOID.

Let AD and BO be the parallel sides of the trapezoid, of
J, which AD IS the less. Let AD b '

/ir

A / by a, BO by h, and the perpendicular distance
'^ / NL of the two sides by h. Draw DE paraUel
j)f to AB, Let G, be the intei-section of

iwi.-^ the diagonals of the parallelogram ABED,
then will G-^ be the centre of gravity of that parall'elo-
gi-am. Bisect CE ia L, join DL, and take DG5=§ DL,
then will G, be the centre of gravity of the triangle DEC.
Draw G.ili and G,M, pei'pendiculars to AD ; then since
AG,=i AE, therefore G,M,=^ FE^iA.! And since
DG, = tDL, therefore G,]V[, = | NL = | A. i Suppose the
whole parallelogram to be collected in its centre of gravity
G„ and the whSe triangle in its centre of gravity Q,. Let
G be the centre of gravity of the whole trapezoid, and draw
GM perpendicular to AD. Then would the whole be sup-
ported by a single force equal to the weight of the trapezoid
acting upwards at G. Therefore (Art. 17),

MG . AB(^="^i; . ABED + GX . CED
Now, ABOD = i A (ft -h S), ABED - ha,

CED -\h q>^a), G.M, = \h, G,M, = -f A,

.■■ MG ■ i A (t(-|-5} = JA. Aa+IA . iA(S~o),

.-. MG (»+5) = Aa+I A (h—a) = \ h (a+25),

.■.MG = iA.-t^^ (19).

80, THE OEBTRE OF QEAVITT OF ANT ..QUADRILATEEAl, FIGUKK.

Draw the diagonals AC and BD of any quadrilateral figm'e
ABCD, and let them intersect in E,
and from the greater of the two parts,
BE and DE, of either diagonal BD set
off a part BE equal to the less part.
Bisect the other diagonal AG in H, join
» HF and take H6 equal to one third of
HE ; then will G be the centre of gi'avity of the whole
figure.

For if not, let g be the centre of gi-avity, join HB and HD
and take HG, = -J HB and HG, = -J HD, then will G, and
Gj be the centres of gravity of the triangles ABC and ADC

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28 THE CEHTRE OF GEATITT.

respectively (^Art, 26). Suppose these triangles to ba col-
lected in their centres of gravity G,, G, ; it is evident that
the centre of ^avity g, of the whole figure, -will be ia the
Btraight line joining the points G-, G, ; let this line interaect
.AC m K ; then since a preeeure eqnal to the weight of the
•whole figiire acting upwards at a, will be in eqnilibrinm with
the weiglits of the triangles collected in G, and G^, we have,
by the principle of the equaUty of momenta (Art 15),

X^ . ABCD=:EG, . ABO— KG, . ABC.

]^ow since HG, =^ J HB, and HG, = i HD, therefore G, G,
is parallel to DB, therefore KG, = i BE, and KG,=:i DE.
ITow let the angle AED = BEC^i. Therefore the perpen-
dicnlai- from B upon AC = BE sin. i, and that from D = DE
sin. I, therefore area of ti-iangle ABC = \ AC , BE sin. j,
and area of triangle ADC ^ J AC , DE sin. j, therefore ai'ea
of quadrilateral ABOD = i AC . BE sin. i+i AC . DE
ein. i = -J (BE+DE) AC sin. I. Substituting these values in
the preceding equation,

K^. I (BE + DE) AC 6ia.i = i BE . I AC . BE sin..—

i DE . 4 AC . DE sin. .,

^ l^(BE+DE) = i (BE^— DE"^,

.■.%=^||^:^=i(BE-DE) = i(BE-BF) = iEE.

But since HG = i HF, .■.KG = i FE, .•.K^ = KG_; that
ie, the tme centime of gravity g coincides with the point G.
Therefore, &c. [q.e.d.]

*31. In the examples hitherto given, the centi'e of pressure
of a system of weights, or theu' centime of gravity, has been
determined by methods which are mtMreet as compared with
the direct and general method indicated in Article 17. That
method supposes, howevei', a determination of the sum of the
momenta oi the weights of all the various elements of the
body in respect to three given planes. Now in a contimiums
body these elements are mjmite in number, each being infi-
iiitely small ; this determination supposes, therefore, the sum-
mation of on infinite number of mfinitely small quantities,
and requires an application of the principles of tJie inti-egal

Let ^M be taken to represent any small element of tlia

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THE CENTEE OF GEAVWr. 29

volume M of a continuous body, and » its perpendicular.
distaiicG from a given plane. Then will ajji aM represent
the moment of the weight of this element ahout that plane,
fi. representing the weight of each un-ii of the volimie M.
Let (J-^fB aM represent the sum of all such moments, taken in
respect to all the small elements, such as ^M, which make
up the volume of the body. Then if G-j represent the dis-
tance of the centre of gravity of the body from the given
plane ; since pSajAM represents the sum of the inomenU of a
system of parallel pressures about that plane, f^M the snm of
tnose pressures, and G, the distance of their centre of pres-
sure ft'om the plane (Art. 19), it follows by equation (18) that
_ l>.Xx . AM _ 2^. AM
''" .U.M ~ Si ^ ''

Now it is proved in the theory of the integral calculuB,*
that a sum, such as is represented by the above expression
2icaM, whose terms are infinite in number, and each the pro-
duct of a finite quantity x, and an infinitely small quantity
aM, and in which M is, as in this case, a function ot ai (and
therefore as a function of M), is equal to the definite integral

/ xd}iL Therefore, generally.

<^..

/

xdM

Similarly,

fydM

■ (31).

In the two last of which equations y and z are taken to repre-
sent, respectively, the distances of the element aM of the

« Poiiaon, Journal de I'Eeole Folytechnfque, ISme cahier, p. 320, or Art. 2,

in the Treatise on Definite Integrals in the Encyclopfedia Mctropolil.ana bj tlw
iiutlior of thia work. See Appendix, note A.

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30 THE CEHTKE OF

body from two other planes, as x represents its dietance from
the first plane ; and G, and G, to represent the distances of
its centre of gravity from those planes. The distances G,,
G„ G„ of the centre of gravity from three different planes
being thus known, its actual position in space is fully deter-
mined. Ihese three planes are usually taken at right angles
to one another, and are then called rectangular co-ordinate
planes, and their common intersections rectangular co-ordi-
nate axes.

If the centre of gravity of the body be known to lie in a
certain plane, and one of the co-ordinate planes spoken of
above, as for instance that from which G^ is measured, be
taken to coincide with this plane in which the centre of grar
vity is known to He, tlien Gj = 0, and the pwition of the cen-
tre of gravity is determined by the two first only of the above
three equations. This case occurs when the body, whose
centre of gravity is to be determined, is «>fm/metrieal about a
certain plane, since then its centre of gravity evidently lies
in its plane of symmetry. If the centre of gravity of the
body be known to lie in a certain Tme, and two of the co-or-
dinate planes, those for instance from which G, and G, are
measured, be taken so as to intersect one another in that line,
then the centre of gravity will be in both those planes ; there-
fore G, = 0 and G^ = 0, and its position is determined by the
first of the preceding equations alone. This case occurs
when the boify. is symmetnoal about a given line ; its centre
of gravity is men manifestly in that Hne.

*33. The centee of geavitt of a cusved line wmcn lies

WHOLLY IN THE SAME PLANE.

Taking M to represent the length 8 of such a line, we
have, by equations (21),

G. =-/^, G, =-^ . . . (22),

Example. — Ldi it he regmred to det&rmine the centre of
yrmity of a cvrcviar arc EF,

The centre of gravity of such an arc is evidently in the
radius CA, which bisects it; since the arc
is symmetrical about that radius. Take a
plane Oy perpendicular to this radius, and
• passing through the centre, to measure the
moments from. Let ee represent the dis-
tance PM of any point P m tins arc from

%.

, Google

OF A CCIIVED HKli- oX

this plane; also let s represent the arc PA, and S the arc
EAF, a the radius CA, aiid C the chord Ei'.

.-. 3! = PJI = OP COS. CPM = CP COS. ACP = a cos. ^

IS is

.\JxdS=:af COS. — -(?s=(s° / COB. — (i{ — l=2((.''sin.i — )'
— JS —IS

the integral being talcen between the limits ^S and — ^,
these are the values of s which correspond to tho
points F and E of the arc,

Kow2ijdn.i (—) = chord of EAF = 0, :.JxdS = aC,
■ G,=^ (23).

The distance of the centre of gravity of a circular arc from
the centre of the circle is therefore a fourth proportional to
the length of the arc, the length of the chord, and the radius
of the arc.

*Zd, The centre of gea.vitt of a citrvilineae aeea

WHICH LIES WHOLLY IN THE SAME PLANE.

Let BAG represent such an area. If a> and y represent
the perpendicular distances PN and PM of any
point P in the curve AB from planes AC and
AD, perpendicular to the plane of the given area
and to one another, and M represent the area
PAM, then, considering this area to be made up
of rectangles parallel to PM, the width of each
. f which is represented hy the exceedingly small quantity
^x, the area ^M of each sucli rectangle will he represented
by 1/i^, and its moment about AD by t^-xy^x.

fT:

Therefore by equation (20), G, = ^^^ = ^— . . (24).

A similar expression determines the value of G, ; but one
more convenient for calculation is obtained, if we consider
the weight of each of the rectangles, whose length is y, to
be collected in its centre of gravity, whose distance from AO

./Google

OF OEAVITT.

ie ^j. Tile moment of tlie weight of eacli rectangle about
AC will then be represented-^^Aj/'Aa:; whence it follows that

G,

a M

. (26).

Example. — Su;ppose the curve APB to he a^arcihola^ whose
axis is AC.

Ey the equation to tlie paraliola y' = 4aie, if a
be the distance of the focus from the vertex,
MoreoveT, the limits between which the integi-al
is to he taken are 0 and x, and 0 and y,, since at
A, ic = 0, y = 0, and at 0, a; = a;,, y = y^,

•lierefore fwydx = 3 |/ ii rxldx=^ \/ im^i; also, M ^=fydx
= 2 j/a Coi^doi = K v'ffla'il, therefore Gi= ^a!;,
Also, lifdie = ^(i / iB(&t=3ffla!'=^;r-jandM=- V iKc,^=^,

therefore G. = -^y,.

If, then, G be the centre of gravity of the parabolic area
ACB, then AH = ^ AC, HG = | OB.

* 34r. The centee op GKAviTr of a suefacb of eevoixition.

Any surface of revolution BAG is evidently Bynimetrical
J, about its axis of revolution AD, its centre of
i^;-^ gravity is therefore in that axis. Let the mo-
"■ mentsbemeasLired from a plane passing through
A and pei-pendicnlar to the axis AD, and let a;
and y be co-ordinates of any point P in the
■■■ generating curve APB of the surface, aud s the
length of the curve AP. Then II being taken to represent
the ai'ea of the surface, and being supposed to be made up
of bands pai-allel to PQ, the area iiM. of each such band is
represented (see Ai-t. 40.)* by '^try^s, and its moment by

* Church's Diff. Calculus, Art. 91.

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OF A atrETACE.

)^Jxy

Example.' — To determme the centre of gravity of the sut'
face of any zone or segment of a sphere.

fLet B,ACi rep^'eeent the surface of a sphere,
■„ whose centime ia D, and whose radius DP ia repre-
sented by o, and the arc AP by s. Then x = DM
, =DP COS. PDM = » COS. % s-^PM^^DP ein.

PDl£=a sin. -, .*. 2iKy = 2a'' sm. _cos.- = (('' sin.—.
a a a a

S, S,

.', 277 / xyds — 77(X° / sin. ~ ds

S, S,

1 , ( 2S, 2S, )

^=^T:a i COS. _i._ COS, i \

= irf|(l + c,».^)-(l + co..^)|

= ™' icOB.'^' - COB." ?!l ..... (27).

13'

where S^ and S^ arc the values of s at the pointa
Bi and Bj, where the zone is supposed to ter-
minate.

■. M = 2-^ I yds

= SiTffl / sin. - ds = 2770.' J COS. -? — cos. -! t ,
<J a \ a a\

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34 THE CENTKE OF GEAYITT

=-i DE, + DE, I =D£ (28),

if E te the bisection of EiE,,

If S, := 0, or the zone commence from A, then

■ (29).

*35. The cehtui

Any solid of revolution BAG is evidently eymmotrical
about its axis of revolution AD, its centre of
gravity is therefore in that Hne ; and taking a
plane passing throngli A and perpendicular to
that axis as the plane from which the moments
are measured, we have only to determine the
distance AG of the centre of gj'avity, from
that plane.

Kow, if jc and f represent the co-ordinates of any point P
in the generating curve, and M the volume of the portion
PAQ of this sond, then, conceiving it to be made up of
cylindrical laminse parallel to PQ, me thickness of each of
which is iia;, the volume of each is represented by '«y''^x, and
its moment by ftMey'^x.

;.&,=

^f^J

M

. (30).

ExAMPi-E, — To determine the centre of gravity of amjsoUd
segment of a sphere.

fLet B,AOi represent any such segment of a
sphere whose centre is D and its radius a. Let x
■u and y represent the co-ordinatas AM and MP of
any point P, x being measured from A ; then by
the equation to the circle j'"— Saic— a^,

Also, M=-f/]/dx = 'xfi^ax-^) dx^x! (aa^^-ii

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OF THE SEGMENT OF AN AECII. dO

:, 0, = — * =ix. . (-^ (31).

If the segment become a hemisphere, x^=a, :. G;=fa.
36. The centre of gravity of the sector of a airole.

Let CAB represent etieh a sector; conceive the arc ADB
to oe a polygon, of an infinite number of sides
^^ and lines, to be drawn from all the angles of the
c<^-aJ-ji> polygon to the centre C of the circle, these will
^^V diTide the sector into as many triangles. Now
'B the centre of gravity of each triangle will he at
a distance from 0 equal to f the line di-awn fi-om the vertex
0 of that triangle to the bisection of its base, that is equal
to f the radius of the circle, so that the centres of gravity of
all the triangles will lie in a circular arc FE, whose centre is
0 and its radios OF equal to ^OA, and the weights of the
tnangles may be supposed to be collected in this arc FE,
and to be uniformly distributed throurfi it, so that the cen-
tre of gravity G of the whole sector CAB is the centre of
gravity of the circular arc FE. Therefore by equation (33),
ff S", C', and a\ represent the arc FE, its chord FE, and its
radius CF, and 8, C, a, the similar are, chord, and radius of
ADB, then CG = " ; but since the arcs AB and FE are

similar, and that a' = -fa, .'. C = fC and S' = |S. Substi-
tating these values in the last equation, we have

CG = |^ (32).

37. The centre of m^tmiy of any portion of a eircular rmg
or of cm a/roh of equcH voti3SOWS.
Let B,CjCjB, represent any such portion of a circular ring
c, whose centre is A, Let a, represent the
J!j---^i radius, and Ci the chord of the arc B,C„ and
A<2^1g..,^„ S, its length, and let a^, 0, similarly represent
g "^ J the radius and chord of the arc B,C„ and S,
^ the length of that arc.
Also let G, represent the centre of gi'avity of the sector
AB,C|, G, that of the sector AB,C„ and G the centre of
gi-avity of the ring. Then

AG, X sect.ABA+AG x m^gBABA= AG, x sect. ABA
Now (by equation 33), AG,=^^, AG,= | ^•,

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36 THE PEOrEETIES

also Bectoi' AB,C,= ^Si«„ sector AB,C,= JSj«j,
.-. img B,C,C,B,= sect- AB,C - sect. AB,C,= iS.o,- JS

.-. i^ • iSA+A5i(SA-S,»,)=l^ . i . Sa,

' .•■ AG . (S,<.,-S,«,)=i (Oa'-Ca"),

.Aa = |'g<2g-<

38. The feopeeties of GrLDmirs.

If I&Ia represent amy flame a/rea, and AB ie any mds, m the
same plane, about which the a/rea is made to
J, rmoVoe, so thai NL is hy this r&ool/ubwn made to
'^ gener<de a soUd of rei)okition, then is the mVuTne
y-^ of this soUd eqvM to thai of a prism whose lose
lij -is NL, a/nd whose height is e^rnl to the length
jT of the path which the centre <f gromty G of the
area NL is made to describe.

For take any rectangular area PESQ in iJL, whose eides
are respectively parallel and pei-pendicnlar to AE, and let
MT be tlie mean distance of the points P and Q, or B and
S, from AE. iNow it is evident that in the revolution of
NL about AB, PQ will describe a snperficial ring.

Suppose this to be represented by QFPK, let M be the

centre of the ring, and let the arc subtended by

^--^T^ the angle QMF at distance unity from M be repre-

'»^\ eented oy i, then the area FQPK equals the sector

'""■^' FQM-the sector KPM=i^Q^x(l-iMFxfi=

Kow the solid ring generated by PPSQ is evidently equal
to the supei'flcial ring generated by PQ, multiplied by the
distance PR. This sohd ring equals therefore i (MT x PQ
xPR) or flxl^xP^Q. Now suppose the ai-ea PESQ
to be exceedingly small, and the whole area NL to be made
up of such exceedingly small areas, and let them be repre-
sented by aj, ffij, OS,, lEC. and their mean distances MT by a?,,
a>„ i»„ &c. then the solid annuli generated by these areas
respectively will (aa we have shown), be represented by
tei^,, flajjff,, tejKj, &c. &c. ; and the sum of these annuh,

./Google

OF GULDIHUe. 37

■ the whole solid, will be represented by Sx^a^+6x,

by ^' . - - -
Sw,a, + &c., or by i{x,a,+!s^a^+a!,a^+&c.). Now if /*■ repre-
sent tlie weight of any superficial element of the plane NL,
iB,«,fi=the moment of the weight of a, about the axie AB,
ai,a3H=tliat of the area a, about the same axie AB, and so
on, therefore the siim (£c,a,+iC,a,+iB,a3+&e.) c.^the moment
of the whole area NL alDOut AB ; but if Q- be the centre of
gj'avity of NL. and GI its distance from AB, then the
moment of NL about AB=GIxNV;
therefore the whole 8olid=J . GI . NL ;
but fl . GI ecLuals the length of the circu-
lar path described by G ; therefore the
volume of the solid equala NL multi-
plied by the length of the path de-
scribed by G, *. e. it equals a pnsm Nil,
whose base is NL, and whose height GH
' is the length of the path deecribed by
G : which is the first prsperty of 6UL-
DINU8.

39. The above proposition is applicable to finding tJie
solid contents of the thread of a screw of variable diame-
ter, or of the material in a spiral staircase ; for it ie
evident that the thread of a screw may be supposed to be
made up of an intinite number of small solids of revolution,
arranged one above another like the steps of a staircase, all
of wMch (contained in one turn of the thread) might be
made to elide along the axis, so that their eurt'acee should all
lie in the same plane ; in which case they would manifestly
form one solid of revolution, such as that whose volume has
been investigated. The principle is moreover applicable to
detennine the volume of any solid (however iri'egular may
be its form otherwise), provided only tliat it may be con-
ceived to be generated by the motion of a given plane area,
perpendicular to a ffl,ven cmwed line, which always pa^es
through the same point in the plane. For it
^^^::^ is evident that whatever point in this curved
line the plane may at any instant be ti-avei--
sing, it may at that instant be conceived to be revolving
about a certain fixed axis, passing through the centre of
cuiwature of the curve at that point ; and thus revolving
about a fixed axie, it is generating for an instant a solid of
revolution about that axis, the volume of which elementary
solid of revolution is equal to the area of the plane multi-

,y Google

plied by the lengtli of the path described by its centre of
gravity ; and this being true of all stich elementary solids,
each being equal to the product of the plane by tlie corres-
ponding elementary path of the centre of gi-avity, it follows
that the whole volume of the solid is equal to the product
of the area by the whole length of the path.

40. If AB represent (my cttrved line made to r&volve about
the axis AD so as to generate the sur-
face of revolution BAG, and G le the
centre of gravity of this cwmed Une,
then is the area o/tki^ swface equal
to the rn'od/uot <f the length of the
curved line AE, Jy the length of the
path described hy the point G, during
the revohition of the curve about AD. Tliis is the seco^id
property of Otdddrms.

Let PQ be any small element of the generating curve,

and PQFK a zone of the surface generated by this element,

this zone may be considered as a portion of the surface of a

cone whose apex is M, where the tangents to the curve at T

and Y, which are the middle points of PQ and FK, meet

when produced. Let this band PQTK of the cone QMF be

developed^, and let PQFK represent its develop-

-^^ ment ; this :Sgure PQFK will evidently be a circu-

<&^^ lar ring, whose centre is M ; since the devolop-

a ment of the whole cone is evidently a circular

sector MQF whose centre M corresponds to the

apex of the cone, and its radius MQ to the side MQ of the

cone.

Now, as was shown in the last proposition, the area of
this circular ring when thus developed, and therefore of the
conical band before it was developed, is represented by
* . MT . PQ, where ^ represents the arc subtended by QMF
at distance unity. Now the arc whose radius is MT is
represented by A . MT ; but this arc, before it was developed
from the cone, formed a complete circle whose radius waa
NT, and therefore its circumference S^NT ; since then the
circle has not altered its length by its development, we
have

* If the cone be supposed covered with a flexible sheet, and a band such
BE PQFK be imagined to be cut upon it, and then unwrapped fram the cona
and laid upon a plane, it ia called the development of tbe band.

, Google

OF QDIDISIJS. )*y

Substituting this value of AMT in the expression for the ares
of the band we have

area of zone PQFK=2* . NT . FQ.
Let the surface be conceived to be divided into an infinite
number of such elementary bands, and let the lengths of
the corresponding elements of the curve AE be represented
by s^, s„ «„ &c. and the corresponding values of WT by y,,
y„ y„ &c. Then will the areas of the corresponding zones
be represented by 3* ^iS,, 2*^585, 2fy/„ &c. and the area of
the whole surface BAG by 2*^,s,+2iri//,+ 2*^,8,+ .... or
by 3''(s',S3+2/i*5+2/A+ -■-■}■ ^'"■^ ^^^°® ^^ the centre of
gravity of the curved line AB, therefore AB . GBi- repre-
sents the moment of the weight of a uniform thread ov wire
of the form of that line about AD, f- being the weight of
each unit in the length of the hne : moreover, this moment
eq^uals the sum of the moments of the weights s,(j.,,g,H', s,M-,
&c. of the elements of the Hue.

/■AB . G-B>=(yA-|-y,s,+S'a3,+ . . . ->

.-.AB . GH=y.s.+yA+i/A+ ■ ■ ■ •

Therefore area of surface BA0=2'n-AB . GH=AB
. (2^GH).

But 2*GH equals tlie length of the circular path describei3
by G in its revolution about AD. Therefore, &c.

This propcffiition, Hke the last, is true not only in respect to
a surface of revolution, but of any surface generated by a
plane curve, which traverse perpendicularly another curve
of any form whatever, and is always intersected by it in the
same point. It is evident, indeed, that the same demonstra-
tion applies to both propositions. It must, however, be ob-
served, that neither proposition applies imless the motion of
the generating plane or curve be such, that no two of its con-
secutive positions intersect or cross one another.

41. The volume of am/ pnmcated prismatio or oyUndrietd
iody ABCD, (^ whwh one etstrmhity CD is perpendioulm'
to the sides m the prism,, and the other AB i/ncUned to
them, is equal to that of cm wpright prism. ABEF, ha/ving
for Us lose the plcme AB, am.d for %ts height the verpeiv-
di<mla/r height GN of the centre of gramity G ofiheplaaie
DO, ahove theplaiu of AB.
For let I represent the inclination of tlio plane DC to AB ;

./Google

THE PEOPEETIES OF GrU)D;US.

take «i, any small element of tlie plane
CD, and let mr be a prism whose base is m
and wiiMe sides are parallel to AD and
BC ; of elementary prisms similar to wliieh
tbe whole solid ABCD may be supposed
to be made up. Now the volume of tliia prism, whose base
is m and lis height nw, equals mrxm = sec. i x (mr . cos. i)
xm = sec. I X (mr . sin. mrm.) m = sec, i x mn x in.

Therefore the whole soHd equals the sum of aU such pro-
ducts as inn x m, each such product being multiplied by tlie
constant qnautity sec. i, or it is equal to the sum just spoken
of, that sum being divided by cos. i. Let this sum be repre-
sented by 2m« X m, therefore the volume of the eohd is re-
presented by . Now suppose CD to represent a,

thin lamina of uniform thickness, the weight of each square
unit of which is (J-, then will the weight of the element m be
repr^ented by f* X m, and its moment about the plane ABN
by fri X frni X m, and f-Sm?! x m will represent the sum of the
moments of all the elements of the lamina similar to m. about
that plane, Now by Art, 15. this sum equals the moment of
the whole weight of the lamina i* x CD supposed to be col-
lected in G, about that plane. Therefore
(J. X CD X !NG=fi2mji x m,
.■. ^D X NG = Snwi X m

Substituting this value of Smn x n, we have
volume of solid = see. i x CD X HG.

But the plane CD is the projection of AB, therefore CD
= ABcos. I, .-, CD X sec, i = AB ;

;. vol. of solid ABCD = AB x NG = vol. of prism ABEF.
Therefore, &c.

[Q. K. D.]

./Google

]?^R T II.

BYKAMICS.

43. MoTiOH JM change of place.

The science of DYNA&nc8 is that which treats of the laws
which govern the motions of material bodies, and of their
relation to the forces whence those motions result.

The SPAOEs described by a moving body are the distances
between the positions which it occupies at different succes-
sive periods of time.

Uniform motion is that in which ec[nal spaces are de-
ecribed in equal successive intervals of time.

The vELociTT of uniform motion is tlie space which a
body moving uniformly describes in each second of time.
Thus if a body move uniformly with a velocity represented
by V, and during a time represented in seconds by T, then
the space S described by it in those T seconds is represented

by TV, or S=TY. Whence it follows that Y = %siiA. T=^'
J ' TV

60 that if a body move uniformly, the space described by it
is equal to the velocity multipUett by the time in seconds,
the velocity is equal to the space divided by die time, and
the time is equal to the space divided by the velocity.

43. It is a law of motion, established from constant obser-
vation upon the motions of the planets, and by experiment
upon the motions of the bodies around us, that when once
communicated to a body, it remains in that body, tmaffected
by the lapse of time, carrying it forward for eve.r with the
same velocity and in the same direction in which it first be-
gan to move, wnless some force aot aft&rwarda in a contrccry
a/ireotion to destroy it*

* This ia the first LiW of uotion. For anmeraos illuatratious of thia fun-
damental law of motion, tie reader la referred to the author's work, entitled;

iLLDSTBmONS OV MCCBAtllCS, Art. 193.

, Google

43 YBLOCITT.

The velocity, at any instant, of a body moving with a
VAEiABLE MOTION, is the space which it would describe in
one second- of time if its motion were from that instant to
become tusifoem.

An AccELffiKATiNG POECE is that which acting continually
upon a body in liie direction of its motion, produces in it a
Continually increasing velocity of motion.

A EETAKonsG FOECE 18 that which acting upon a body in
a direction opposite to that of its motion produces in it a
(;ontinually diminishing velocity.

An iMFTJLsrvE FOECE is that which having communicated
motion to a body, ceases to act upon it after an exceedingly
small time from the commencement of flie jnotion.

44. A UMITOEMLY accelerating or retarding force is that
which produces equal increments or decrements of velocity
in equal successive intervale of time. If f represent the
additional velocity communicated to a body by a uniformly
g force in each snecessive second of time, and T

the number of seconds during which it moves, then since hy
the first law of motion it retains all these increments of velo-
city {if its motion be unopposed), it follows that after T
seconds, an additional velocity represented by /'T, will have
been comnranicated to it ; and if at the com/m&nGem&nt of
this T seconds its velocity in the same direction was Y, then
this initial velocity having been retained (by the first law of
motion), its whole velocity wUl have become V-f-/T.

If, on the contrary, / represents the velocity continnally
taken oAoay from a body in each successive second of time,
by a unifoimly retai'ding force, and V the velocity with
which it began to move xn a direction opposite to that in
which this retarding force acts, then will its remaining velo-
city after T seconds be represented by V— /T; so that gene-
rally the velocity V of a body acted upon by a uniformly
accelerating or retarding force is represented, after T seconds,
by the formula

v=T±/T (34).

Tile force of gravity is, in respect to the descent of bodies
near the earth s surface, a constantly accelerating force,
increasing the velocity of their descent hy 82f feet m each
successive second, and if they be projected upwards it is a
constantly retarding force, diminishmg their velocity by that
quantity in each second. The symbol^ is commonly used to

./Google

represent this number 32f ; so that in respect to gi'avity the
above formula becomes ii=Y±/r, the sign ± being taken
according as the body is projected downwards or upwards,

A TAEiAKLE acoeleratmg force is that which commtinieates
unequal increments of veEocity in eqaaX successive intervals
of time ; and a variable rettwdvng force that which talies
away unequal decrements of velocity.*

45. To DETI'ISMIKE THE EEIATTON BETWEEN TOE VELOOrTY A3D
THE SPAOB, ASD THE SPACE AND TIME OF A BODy's MOTION.

Let AM,, MiM,, M,M„ &c. represent the exceeding small
successive penods of a body's motion, and
AP the velocity with which it began to
move, M,Pi the velocity at the expiration
of the first interval of time, MjPjthat at
the expiration of the second, MjP, of the
third interval of time, and so on; and
instead of tbe body varying tbe velocity of its motion corb-
tinually throughout tlie period AM,, suppose it to move
through that interval with a velocity wliich is a mean
between the velocity AP at A, and that MjP, at M„ or with
a velocity equal to |(AP+M,P,).

Since on this supposition it mov^ with a 'uniform motion,
the space it desciibes dui'ing the period AM, equals the
product of that velocity by that penod of time, or it equals
|-(AP+M,P,)AM,. Now this product represents the area
of the ti-apezoid AM,PiP, The space described then in the
interval AM,, on the suppc^ition that the body moves during
that interval with a velocity which is the mean between
those actually acquired at tlie commencement and termi-
nation of the interval, is represented by the trapezoidal
areaAM,P,P.

Similarly the areas PjM^, PjM„ &c. represent the spaces
the body is made to describe m tlie successive intervals
M,M„ MjM„ &c. ; and therefore the wliole polygonal area
APCB represents the whole space the body is made to
describe in the whole time AB, on the supposition that it
moves in each successively exceeding small interval of time
with the mean velocity of that interval. Now the lees the
intervals are, the more nearly does this mean velocity of each
interval approach the actual velocity of that interval ; and
if they be lufinitely emaU, and therefore infinitely great in

* Note (i) Ed. App.

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a MOTION TJKIFORM.lt

number, then the mean velocity coincides with the actual
velocity of each interval, and in tliia case the polygonal area
paeses into the cm-vilinear ai-ea APCB.

Generally, therefore, if we represent by the abaeissm of a,
curve the times through which a body has moved, and bv
the corresponding ordinates of that curve the velocities wMcn
it has acquired after those times, then the a/rea of that curve
will represent the spaee throi^h which the body has moved ;
or in other words, it a curve PO be taJien such that the num-
ber of equal parts in any one of its absciesfe AM, being taken
to represent the number of seconds during which a body has
moved, the number of those equal parts in the corresponding
ordinate MjPs will represent me number of feet in the velo-
city then acquired; then the space which the body has
described will be represented by the number of these equal
parts squai'ed which ai-e contained in the ai'ea of that curve.

46. To DJiTEItMlNE 'XnE SPACE DEgCEIEED IN A GIVEN TIME BT
A BODT WHICH :B PHOJECTED WITH A GIVEN VELOCITY, AND
WHOSE MOTION IB UNIEOEMLY ACCELERATED, OR TJXUi'ORMLY
RETARDED,

Take any straight line AB to represent the whole time T,
e in seconds, of the body's motion, and draw AD
D*^^^^^B perpendicular to it, representing on the same
pi-^-J^ scale its velocity at the commencement of its
I jM 1 motion. Draw DE parallel to AB, and accord-
ing ae the motion is accelerated or retarded
draw DC or DE inclined to DE, at an angle whose tang'ent
equals/, the constant increment or decrement of the body's
velocity. Then if any'abscissa AM be taken to represent a
number of seconds t during which the body has moved, the
corresponding ordinate MP or MQ will represent the velocity
then acquired by it, according as its motion is accelerated o°r
retarded. For PR = EQ=iDE tan. PDE=AM tan. PDE;
but AM = *, and tan. PDE=/: therefore PE-EQ=/i;.
Also IIM=AD=V, therefoi-e MP=EiI-hFE=Y-h/5i, and
MQ=EM— EQ=V— /i! ; therefore by equation (34), MP or
MQ represents the velocity after the time AM accoi-ding as
the motion is accelerated or retarded. The same being true
of every other time, it follows, by the last propositiou, that
the whole space described in the time T or AB is repr^ented
by the area ABCD if the motion be accelerated, and by the
area ABED if it be retarded.

./Google

ACCELEltATED OK Bl^TAllDED. 43

Now area ABCD=^AB(AD + BO), but AB=T, Ar)=Y,
BO=V+/T,

.-. areaABOD=^T{V+V+/Tj=yT+i/P.

Also ai-ea ABED^^AB (AD+BI'), where AB and AD
liave the same values as before, and BF=Y— /'T,

;. area ABFD=iT(V+V-/T)=:TT-i/r.

Therefore generally, if S represent the space described aftee
T seconds,

8=VT±i/T= (35);

in which formula the sign ± w to he taken according as tlie
motion is accelerated or retarded.

4T. To nSTEltMlNE A SELA'nOK BETWEEN THE SPACE DESGEIBED
AND THE VEI.OCrrY ACQDIEiED BT A BODY "WHICH 18 I'KOJECTED
WITU A GIVEM VELOCITY, AND ^V^OSE MOTION IS TNIFOEMLT
ACCELEEATED OB KETAEDED,

Let V bo the velocity acquired after T seconds, then by
equation (34), v = Y ±/T, ..-. T=: ±^-^— "'■

p^_.,^D Now area ABCr)=iAB(AD + BC), where
"r^^^H^ AB=T=^-"^' AD=T, B0=?),

. areaABOD^i^^^y-''(Y+t>)

.i- ^-i! \ A-1\\ — i-^ -Jl

f "-'^'!-^^r

aiea ABFD = iAB(AD+Br), whore AB=T =

fe.=S,

/
AD=V, BF=».

.-. ™.ABFD=-ifc:I)j'^= _i(jL~p.

Therefore generally, if S represent the space tlirough which
M—Y")
the velocity d is acquired, then S = ±i--^ — 3 — -^

.■.:i,»— -V=-t2/S (36);

in which formula the rh sign is to be taken according as
the motion is accelerated or retarded.

If the body's motion be retarded, its velocity v will eventu-
ally be destroyed. Let Si he the space which will have been

./Google

4b THE CSIT OF "WORK.

deseri"bed when v thus vanishes, then "by the last eqaation
0-V'= - 2/S,

.-. ^ = 2/8, (37),

■where V is the velocity with wliieh the hody is pi'ojeeted
in a direction opposite to the force, and S, the whole space
which by this velocity of projection it can be made to
describe.

K the body's motion be accel&rated, and it fall from Test,
or have no velocity of projection, then 'o'— 0 = +2/'S,
.-. 'y'=2/S (38).

Let Sj be the space through which it must in this case
move to acquire a velocity v equal to that with which it
was projected in the last case, therefore V°= ^Sj. "Whence
it foUowe that Si=8j, or that the whole space S, through
which a body will move when projected with a given velo-
city Y, and uniformly reUwAed by any force, i8 equal to the
epace S„ through wMch it must move to acquire that velo-
ci\j when unifomdy accelerated by the same force.

in the case of bodies moving freely, and acted npon by
gravity, f equals 82^- feet and is represented by g ; and the
space S„ through which any given velocity V is acquired, is
then said to be that d/ue to that velocity.

"WOEK.

48. "WosK is tie union of a continued pressure with a
continued motion. And a mechanical agent ie thus said to
woEK when a preesm-e is continually overcome, and a point
(to which that pressure ie appUed) continually moved by it.
Neither pressure Uor motion alone ie sufficient to constitute
vxrrJc ; so that a man who merely supports a load upon his
shouldei'B, without moving it, no more works, in the sense in
which that term is here used, than does a column which sus-
tains a heavy weight upon its summit ; and a stone, as ij. falls
freely in vaeuo, no more works than do the planets as they
wheel unresisted through space.*

49. The uNrr of work. — ^The unit of work used in this
country, in terms of which to estimate every other amount

./Google

KK. 47

of work, is the woi-k neeeesaiy to overcome a pressure of one
pound tlu-ough a distance of one foot, in a direction opp()site
to tiiat in wliich a pressure acts. Thus, for instance, if a
pound weight be raised through a vertical height of one foot,
one unit of work is done ; for a pressure of one pound is
overcome through a distance of one foot, in a dii-ection oppo-
site to that in which tlie pressure acts.

50. The mmber of tmita of work vsoessary to overcome a
presswpe of M mvmdK thfrough a, dist<moe of N fiet, ia
e^ial to the^rdauot MR.

For since, to overcome a pressure of one pound through
one foot requires one unit of work, it is evident that to over-
come a pressure of M pounds through thi. same distance of
one foot, will rec[uire M units. Since, then, M units of work
are required to overcome this pressure through one foot, it
it evident that K times as many units {i. e. Nltf) are required
to overcome it through N feet. Thus, if we take U to repre-
sent the number of imits of work done in overcoming a con-
stant pressure of M. pounds through N feet, we have
tr=MK (39).*

51. To ESTIMA-ffi TnE WOKK DONE UHDER A VAKIABLE

Let PC be a curved line and AB its axis, such that any-
one of its abscisBEe AM,, containing as many
"y equal parts as there are units in the space
through which any portion of the work has
a been done, the corresponding ordinate M,P,
may contain as many of those equal parts,
(IS there are in the pressure under which it is then being
done. Divide AB into exceedingly small eoual parts, AM,,
iliMj, &c., and draw the ordinatee M,Pi, M^Pj, &c. ; then if
we conceive tlie work done thi-ongh the space AM, (which
is in reality done under pressures varying ii-om AP to M,P ),
to be done uniformly under a pressure, which is the arith-
metic mean between AP and M,Pj, it is evident f]iat the
number of units in the work done through that small space
will equal the number of sqiiare units in the trapezoid
APPiMi (see Art. 45.), and similarly with the other trape-

" Xote (k) Ed. App,

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48 THE EESOLOTIOS

eoids ; so that the number of units in the whole work done
tliroiagh the space AB will equal the number of square units
in the whole polygonal area APP,P^„ &c., CB.

But since AM,, M,Mj, &c., are exceedingly small, thia
])olygonal ai-ea passes into the curvilinear area APCB ; the
whofi work done is therefore represented by the number of
square equal parts in this area.

How, generally, the area of any curve is represented by

the integral /j/ifej where y represents the ordinate, and x

the corresponding abscissa. But in this case the variable

SresBure F is represented by the ordinate, and the space S
escribed under this variable pressure by the abscissa. If
tbei'efore IT represent the work done between the values S,
and 8j of S, we have.

S,
V-^TdS (40).

Mecm pressure is that under which the same work would
be done over the same space, provided that pressm-e, instead
of varying throughout that space, remained
the same : thus, the mean pressure in re-
spect to an amount of work represented by
the curvilinear area AEFC, is that under
which an amomit of work would be done
by the rectilineal area ABDO, the area ABDO
being equal to the curvilinear area AEFO ; the mean pres-
sure m this case is represented by AB. Thus, to determine
the mean pressure in any case of variable pressm'e, we have
only to find a curvilinear area representing the work done
under that variable pressure, and then to describe a rectan-
gular parallelogram on the same base AC, which shall have
an area equal to the curvilinear area.

If 8 represent the space described under a variable pres-
sure, TJ the work done, and p the mean pressure, tlien

^S = TJ, therefore p = -^ .*

52, To estimate the work of a pressure, whose direction is not
that m whieh itspomt of applieation is made to -move.

Hitherto the work of a force has been estimated only ou

" Note (I) Ed. App.

./Google

the supposition that the point of applica-
tion of liat force is moved in the direction
in which the force operates, or in the oppo-
site direction. Let PQ be the direction of
a pre^ure, whose point of application Q
is made to move m the direction of the
straight line AB. Snppose the pressure P to remain con-
stant, and its direction to continue parallel to itself. _ It is
required to estimate the work done, whilst the point of
application has heen moved from A to Q.

Resolve P into E and S, of which E is parallel and S per-
pendicular to AB. Then since no motion takes place in the
direction of SQ, the pr^sure S does no work, and the whole
work is done by E; therefore the work = R . AQ.

Wow E=P . cos. PQK, therefore the work =P . AQ cos.
PQE. Prom the point A draw AM perpendicular to PQ,
thenAQeos. PQE=QM; therefore work=P . QM. There-
fore the work of any pressure as above, not acting in the
direction of the motion of the point of application of that
pressure, is the same as it would have been if the point of
application had been made to move in the direction of the
pressure, provided that the space through which it was eo
moved had been the projection of the space through which
it actually moves. The product P . QM may be called the
work of P resolved in the direction of P.

Tht! above proposition which has been proved, whatever
may be the dist^ince through which the point of application
is mo'v ed, m that particular case only in which the pressTu-e
remains the same m amount and always parallel to itself, is
evidently tine foi exceedingly small spaces of motion, even
if the piessuie be variable both in amount and direction;
since for such exceedingly small variations in the positions
of the points of application, the variations of the pressures
tliemselves, both in amount and direction, arising from these
vai'iations of position, must be exceedingly small, and there-
fore the resulting vaiiations in the woi'k exceedingly small
as compared with the whole work.*

• Note (m) Ed. App.

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THE WOES OF

1. If o/nA/ nuntber of pressures P„ P,, P„ he appUed to the
same po^mt A, am,d r&mavn, Gonstant and pa/raUel to thrnnr

selvm, whilst the poi/nt A is jnade to move through the
straigM line AB, then the whole wo'rh done is equal to the
Slim of the wor'ks of the different messwes resolmed im. the
directions of those presswes, eachieing taken negatively
whose point ofwpphcation is THade to move in am- opposite
direction to the pressure vpon it.

:c. represent the inclinations of the pres-
sures P„ P,, &c. to tho line AB, then ■will
the resolved parts of these pressures in the
direction of that line be P, cos. o-i, P, cos.
a„ P, COS. «j, &c. and they will be equiTa-
lent to ft single pressure in the direction
of that line represented by P, cos. "-,+?,
COS. «5+Ps COS. a„ &c. in which sum all
those tei-nis are to be taken negatirely which involve pres-
sures whoso direction is from B towards A (since the single
pressure from A towards B is manifestly equal to the difter-
enee between the stun of those resolvea pressures which act
in that direction, and those in the opposite direction). There-
fore the whole work is equal-to )Pi cos. a, + Pj cos. a^ -f- P,

COBS-H }. AB = P. ■ AB COS. a. + P, . AB cos, n,

+P,AB COS. «,+ ... =P, . BM, + P, . B,M+ P, . BM,-|-

; in wMch expression the successive terms are the

works of the different pressures resolved in the several
directions of those pressures, each being taken positively or
negatively, according as the direction of the corresponding
pressore is towards me direction of the motion or opposite
to it.

Thne if U represent the whole work and Ui and U, the
snms of those done in opposite directions, then
U=U,-U,» (41).

54. ^ any number of pressures (w^ied to ajpovnt le in eoW-
Uffriwn,, and th&i/r point (f appmcatvon he moved, thewhole
worle done iv these pressures -m the d^ection of the motion
win equal the whole work done in the opposite direction.

For if the pressures P„ P„ P„ &c. (Art. 53) be in equi-
librium, then the sums of their resolved pressures in opposite

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directions along AB will he equal (Art. 10) ; therefore tlie
whole work U along AB, whicb by the laat proposition is
equal to the work ot a pressure represented by the dWerenc6
of these snms, will equal nothing, therefore 0 = U,— TJ„
therefore Uj=1j„ that is, the whole work done in one direc-
tion along AB, by the pressures ?„ P,, &c. is equal to the
whole work done in the opposite direction.

55. ^ a tody he acted upon hy a force whose c
amams towards a certavn pomt S, called a eenire of force,
and he made to deserve (my given cwve PA m a mrection
imposed to tJie aotion of that force, mid %> he meaawed on
&A efual to SP, then will the work dme m movmg tlis
hody through the eurve PA he eqital to that which would
he necessa/ry to move it in a straight line from p to A.

Tor suppose the curve PA to be a portion of a polygon of
an infinite number of sides, PP„ P,P„ &c.
Through the points P, P„ P„ &c. describe circu-
lar arcs with the radii SP, SP„ SP„ &c. and let
them intersect 8A in p, p„ p,, &c. Then since
PP, is exceedingly smalt, the force may he consi-
dered to act throughout this space always in a
direction parallel to SP, ; therefore the work done
through PP, is equal to the work which must be
love the body throu^ the distance wtP, (Art. 52.),
, is the j^<fjection of FP, upon the direction SP, of
But mP,-=pp, ■ therefore the work done through
I to that which would be required to move l£e
body along the line 8 A through the distance^, ; and simi-
larly the work done through P,P, is equal to that which
must be done to move the body through p^p,, so that
the work through PP, is equal to that through pp,, and so
of all other points in the curve, Therefore the woii through
PA is equal to that through pA.* Therefore, &c. [q.e.d.]

* Of course it is in this proposition supposed that tlie force, if it be not
constmit, ia dependant for its amount only on the distance of the point at
wMch it acts from the centre of force S; so tliat the distances of p and F
from S b«in^ the sime, the force at p is equal to that at P; similarly tha
force atpi is equal to that at Pi, the force atpj equal to that at P5, &o.

, Google

52 THE WOKK OF

56. If ^ M at an exceedingly great distance as compareA
with Ar , thj&n, aU the Unes djramnfrom S to AP may Se con-
ddered paralUl. This iB the case with the force of gravity
at the surface of the earthj which tends towards a pomt, the
earth's centre, situated at an exceedingly great distance, as
compared ■with any of the distances through which the work
of mechanical agents is usually estimated.

Thus then it follows that the worlc necessary to move a
heavy hody wp any curve PA, or inclined plane, is the same
as would be necessary to raise it in a vertical line_pA to the
same height.

The dimensions of the hody are here supposed to he ex-
ceeding small. If it he of considerahle diinenaions, then
whatever be the height through which its centre of gravity
is raised along the curve, the work expended is the same
(Art 60.) as though the centre of gravi^ were raised verti-
cally to that height.*

57. In the preceding propositions the work has been esti-
mated on the suppoUtion that the body is made to move so
as to increase its distance from the centre S, or in a direction
opposed to that of the force impelling it towards S. It is
evident, nevertheless that the work would have been precisely
the same, if instead of the body moving ^Votti P to A it had
moved from A to P, provided only mat in this last case
there were applied to it at every pomt such a force as would
prevent its motion from being accelerated by the force con-
tinually impelling it towards 8 ; for it is evident that to pre-
vent thk acceleration, there must continually be applied to
the body a force in a direction /totw S equal to that by which
it is attracted towards it ; and the work of such a force is
manifestly the same, provided the ^>ath be the same, whether
the body move in one direction or the other along that patli,
being in the two eases the work of the same force over the
same space, but in oppceite directions.

* The oiih/ force acting upon the body is !n tliia proposition supposed fo be
tliat acting (awards S. No account ia taken of friction oc any other foroea
which oppose themaelyes to its motion.

, Google

58, If there he any number af^M/foUel jiressiires, P„ P„ P„
&c. whose pomts of a^pt/iGOtion are trainsferred, each
through am/ gwen amtOAics from, one position to (mother,
then «s the work which would he neoessa/ry to trcmsfer their
resultamt tJmmgh a space equal to that hy which their
centre of pressure ia displaced in this ohmtge of position,
equal to the di^erence between the aggregate work of those
pressures whose points of appUcation hme hem. moved im
the directions m whdoh thepresswea appUed to them act,
amd those whose points of appUcaUonhaioe heen moved i/n

For (Art, IT.), if y„ %, y^, &c. represent the distances of
the pointe of application of these presBnres from any given
plane in their first position, and h the distance of their centre
of pressure from that plane, and if T„ T,, T, &c. and H re-
present the corresponding distances in the second position,
and if P„ P„ P„ fee. be taken positively or negatively ac-
cording as then' directions are from or towa/rds the given
plane, A {P,+P, + P,+ . . . \ =P,y,+P,y,+P,y, ....

andHfP,+P, + P,+ ,. . . . }=P,Y,+P,T,+P,y,+

.-. [Jl-h] |P,+P,+P,+ . . . } = P, (Y,-y,)+P= (Y -y,)
+7,{Y,-y,)+ (42);

in the second member of which equation the several terms
are evidently positive or negative, according as the pre^ure
P corresponding to each, and the difference Y—y of its dis-
tances &om the plane in its two positions, have the same or
contrary signs. Now by supposition P is positive or ni^ative
according as it acts/?'om or towairds the plane ; also X—y ia
evidently positive or negative according as the point of appli-
cation ot P is moved from or towards the plane ; each term
is therefore positive or negative, according as the correspond-
ing point of application is transferred in a direction tmoairds
that in which its applied pressure acts, or in the opposite
direction.

Now the plane from which the distances of the points of
application are measured may be am) plane whatever. Let
it be a plane perpendicular to the du'eetions of the pressiires,

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THE WORK OF

.X

Let As«/ represent tliis plane, and let P
P' represent the two positions of tlie point
of application of the pressure P (the path
described by it between these two positions
having been any whatever). Let MP and
t M'P' represent the perpendicular dis-
tances of the points P and P' from the
plane, and draw Pm from P perpendicular

to M'P'. Th_en_P (T-3/)=P(MT'-MP}=P . m.P'; but by
Art. 55., P . mP' equals the work of Pas its point of applica-
tion is transferred from P to P'. Thus each term of the second
member of equation (42) represents the work of the corre-
sponding pressure, so that if sm^, represent the aggregate
work of those pressures whose points of application are trans-
ferred towards the directions in which the pressures act, and
^11, the work of those whose points of application are moved
opposite to the directions in which they severally act, then
the second member of the equation is represented by ^m^ —
2^5- Moreover the hrst member of the equation is evidently
the work necessary to transfer the resultant pressure P, 4-
P,-|-P, &c. through the distance H— A, wMmi ie that by
which the centre of pressure is removed yrow* or towards the
given plane, so that if TJ represent the quantity of work
necessary to make this transfer of the centre of pressure,
TJ=2w,— 2m, (43).

59. If the sum of those pai'allel pressures whose tendency
is in one direction equal the sum of those whose tendency

is in the opposite direction, then P^-l-Pj-i- Pj 4- =0.

In this case, therefore, TJ=0, therefore l«,— 2«^=iO, there-
fore 2t(.j=: Sm^ ; so that wAen Ml any s;

3 of thme.'\x>}Mse, ^ovnts of a
% the directions of the pressure smt
I to the aggregate work of those whose
we inoved w. the opposite direetiona.
This case manifestly obtains when tiie parallel pr^sures
are in EQun^rBEiUM, the sum of those whose tendency is in
one direction then equalling the sum of those whose tendency
is in the opposite direction, since otherwise, when applied to
a point, these pressures could not be in equilibrium about
that point (Art. 8.1.

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tCEB. 55

60. The precediEg proposition is manifestly true in respect
to a system of weights, these being pressures whose directiona
are always parallel, wherever their pointe of application may
he moved. !N"ow the centre of pressure of a system of
weights is its centre of gravity (Art. 19). Thus then it fol-
lows, that if the weights composing such a system be sepa-
rately moved in any dh'ections whatever, and through any
distances whatever, then the difference between the aggre-
gate work done wpwards in making this change of relative
position and that done downwa/rm is equal to the work
necessary to raise the sum of all the weights through a height
equal to that through which their centre of gi-avity is raised
or depre^ed.* Moreover that if such a system of weights
be supported in equilibrium by the resistance of any fixed
point or points, and he put in motion, tbTu (since the work
of the resistance of each such point is notJJmg) the a '■'

* This proposilJon haa numeroas applieatioos. If, for instanoe, it be re([mred
to determine flje aggregate expenditure of work in ridsiog the diflerent ele-
ments of ft Btrnoture, its stone, cement, &o., to the different poaitiona they
occupy in it, we mate this calculation bj detenuinine; the work requisite to
raise the whole weight of material at once to the height of the centre of gra-
vity of the structure. If these materials hftTe been carried up by labourers, and
we ace de^roue to inolnde lie whole of their labour in the calculation, we
asoertiun the probable amount of each load, and conceive the weight of a la-
bourer to bo added to each load, and then all these at once to be raised to the
height of the centre of gravity.

Ag^, if it be required to determkie the expenditure of work made in rais-
ing ftie materioi eioavated from a well, or in pumping the water out of it, we
know that (negleotii^ the effieet of friction, and the weight and rigidity of tha
cord) this expenditure of work is the same as though Qie whole material had
been raised at one lift from the centre of gravity of the shaft to the surface.
Let US take another application of this principle which offers so many pracdol
results. The material of a raQwfty eieavation of considerable length is to be
removed so as to form an embankment across a valley at some distance, and it
is required to determine the expenditure of work made in this transfer of th^
material. Here each load of material is made to traverse a different distance,
a resistance from the friction, Ac., of the road being continually opposed to \-i
motion. These re^atancea on the different loads ooosUtutfi a system of para'-
lel pressures, each of whose poinls of applieataon ia separately transferred fro i
one given point to another given point, the directions of tranafer being als )
parallel. Kow by Ihe preceding propoaition, the expenditure of work in a!i
these separate transfers is the same as it would have been had a preaaure equ il
to the sum. of all theae pressures been at once transferred from the centre Oi'
reastance of the excavation to the centre of reaatanoe of the embanliment.
ITow the resistances of the parls of the mass moved are the frictions of its ele-
ments upon the road, and these frictions are proportional ij) the lesightt of the
elemenls ; their centre of reastanoe coincides therefore with the centre of gra-
vity of the mass, and it follows that the espenditnre of work is the same aa
though all the material had been moved at una from the centre of gravity of
the excavation to that of the embankment. To allow for the weight of tht
carriages, as many times the weight of a carriage must be added to the weight
of the material as there are journeys made.

, Google

5H STABiLirr of the cektee.

work of those weights which are made to descend, is equal
to that of those which ai'e made to ascend,

61, ^ a pl(me he taixnperpendiaula/r to the directions of <my
fmmber of pwraUA ;pTesm.vFes and, there he two dAff&rentpo-
sitions oj the pmita of a^Uoation of certam qf these _pi'es~
tt diffffr&nt dietwnces from the
apjmoation of the rest of these
viie distanoe from that plame,
iifim, lothposiMons the system le in equiUhriitm, then
the G&ntre of j)resswe of thefrst mentioned presswes will
he at the same distan^oe from the plane im, both positions.

For since in both positions the 8;™teni is in eCLuilibrium,
therefore in bothpoeitions P,+p5+'r,+ . . . =0,

Now let P^ be any one of the pressures whose points of appli-
cation is at the same distance from the given plane in both
positions,

.-. Y„=2/,, and Y,— y„ = 0,
.-.(Y -2/,)P,+(Y -2/,)P, + - ■ ■ + (Y„_ -2/„_,)F„_.=0,
.-. Y,P,+Y,P, +. . . +Y_,P„_,=j'.P,+2/,P,+ ...+y _,?,_.,
■ Y,F,+Y,F,+ . ■ .+Y„_,F^,_y,F,+y,P,+ . . . +y„_.F„ „
P +P,+ . . , +P,^, F, + P,+ . . . +P,_,

where H^., represents the distance of the centre of pressure
of P„ P, . , . P,_„fi-om the given plane in the first position,
and A^_, its distance in the second position. Its distance in
the fli-st position is therefore the same as in the second.
Therefore, &c.

Fi-om this proposition, it follows that if a system of weif/hts
be supported by the resistances of one or more fixed points,
and if there be any two positions whatever of the weights in
both of which they are in equilibrimn with the resistances
of those points, then the height of the common centre of

favity of the weights is the same in both positions. And
at if there be a series of positions in all of which the
weights are in equilibrium about such a resisting point or
points, then the centre of gravity remains continually at the
same height as the system passes through this series of posi-
tions.
If all these positions of equilibrium be infinitely near to

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WOEE OF PKES8UEE8. 57

one another, tlien it is only dming an infinitely small motion
of the points of application that the centre of gravity ceasea
to ascend or descend; and, conversely, if for an iniinitely
email motion of the points of application the centre of
gravity ceases to ascend or deaeend, then in two or more
positions of the points of application of the system, infi-
nitely near to one another, it is in equilihrium.

WoKK OF PkESSTTKHS APPLIED IN DIlfFEEENT DlKI<:CTIONS TO

A Body moveable about a fixed Aslis.

For let AB represent tie dii-ection of a pressure applied
to a body moveable about a flxedTaxis
' 0 ; the work done by this pressure
■will be the same whether it be ap-

Elied at A or B. For conceive lie
ody to revolve about 0, through an
exceedin^y small angle A 00, or
BOD, so that the points A and I) may describe circular arcs
AC and BD. Draw Cm, Dn, and OE, perpendiculars to
AB, then if P represent the pressure applied to AB, P . Am»
will represent the work done by P when applied at A (Art.
63.), and P . Bn will represent the work done by P when
applied at B ; thei'efore the work done by P at A is the same
as that done by P at B, if km is equal to Bw.

Now AC and BD being exceedingly small, "they may be
conceived to be straight lines. Since BD and BE ai'e
r^pectively perpendicular to OB and OE, therefore /DBE
=^ Z BOE ; * and because AC and AE are perpendicular to
OA and OE, therefore Z OAE = / AOE. Now Am =

ri A

OA . cos. CAE = CA . cos. AOE = ^ . OA . cos. AOE

= — ^ X OE. Similarly Bw = DB cos. DBE:=DB . cos.

BOE =^ 7TTT' ^^ '■'^*' -'-''^^ — (\Tt '

* It 19 a well-knowQ principle of Geometry, that if two lineE be inclinca aX
any angle, imd any two others be drawn perpendicular to these, then the indi^
cation of the la^t two to one another sh^ equal that of the hrat two.

, Google

THE ACODMCLATIOK OF WOEK.

OA' ""*^ Bn = OE j^. But ox^o^-, since the / AOC=
/BOD, therefore Am = B?*,.*

63. If amy numbm- of pressures oe in equilibrium about a
Jkeed aids, th&n. th» lahde work of those which tend to move
the system in one di/reoldon about that axis is egual to the
whole work of those wMoh tend to move it in ike opposite
Section, about the same ams. '¥or let P be any oae of such
a system of pressures, and 0 a fixed axis, and OM perpen-
diculai' to the direction of P, then whatever may be the
point of application of P, the work of that pressure is the
same as though it were applied at M. Suppose the whole
_ system to be moved through an exceeding small
/ angle fl about the point O, and let OM be repre-

"jC sented by p, then will j)fl represent the space

' ~"^~,, described by the point M, which will be actually
"^'' in the direction of the force P, therefore the work
of P=P . » . S. Kow let P„ P„ P„ &c. represent those
pressiu-es wliich act in the direction of the motion, and P'„
P'j, &c. those which act in the opposite direction, and let
p„p„Ps, &c. be the perpendiculars on the first, and^'„ p'„
p'„ &c. be the perpendiculare on the second; therefore by
the principle of the equality of moments P^^,^-P^,+p5p,
+&C. =P'j»', + P'^',+ P'j*'s+&c, ; therefore multiplying
both sides by fl, P^,S + Pj?,fl + P^,fl = P>',fl + P'y,3 +
P'jPV + &c. ; but Pj),fl, P'J>'l^ &c. are the works of the
forces P„ P'„ &c. ; uierefore the aggregate work of those
which tend to move the system in one direction is eqiial to
the aggi'egate of those which tend to move it in the opposite
direction.

6i, The AccuMULAnoN of wobk ih a movihg body.

In every moving body there is accumulated, by the action
of the forces whence its motion has resulted, a certain

amount of power which it reproduces upon any r

opposed to its motion, and which is measm-ed by the work
done by it upon that obstacle. Not to multiply terms, we
shah speak of tliis accumidated power of working, thus
measured by the work it is capable of producing, as accu-
mulated WOEK. It is in this sense that in a ball fired from
* Now (o) Ed. App.

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WOEK. 59

a cannon there ie nndei"atood to be accumulated the work it
reproduces upon the obetaclee which it encounters in its
flight ; that in the water -which flows through the channel
of a miU is accumulated the work which it jielde up to the
wheel;* and that in the cari'iage which is allowed rapidly
to descend a hill is accumulated the work which caiiies it a
considei-ahle distance up the next lull. It is when the j
sure nnder which any work is done, exceeds the resi
opposed to it, that the work is Uius acGwmulaied in a n
body ; and it will subseq^uently be shown (Art, 69.) that in
every case the work accumulated is precisely equal to the
wort done upon the body beyond that necessary to over-
come the resistances opposed to its motion, a principle
which might almost indeed be assumed as in itself evident.

65, The amount of work thus accumulated in a body
moving with a given velocity, is evidently the same, wbat-
<.;vor may have been the circumstances under which its
velocity has been acquired. Whether the velocity of a ball
lias heen communicated by projection from a steam gun, or
explosion from a cannon, or oy being allowed to fall freely
from a sufficient height, it matters not to the r^ult; pro-
vided the same vdoetty be communicated to it in all three
cases, and it be of the same weight, the work aooimmlaied
in it, estimated by the effect it is capable of producing, is
evidently the same.

In like manner, the whole amount of work "which it is
capable of yielding to overcome any resistance is the same,
whatever may be the nature of that r

66. To E3'

I.ATED IN I

Let w be the weight of the body in pounds, and v its
velocity in feet.

!Now suppose the body to be projected with the velocity u
in a direction opposite to gravity, it will ascend to the height
h from which it must have fallen, to acquire that same velo-
city V (Art. 47.) ; there must then at the instant of proj ectioa
have been accumulated in it an amount of woi-k sufficient to
raise it to this height h ; but the number of units of work

./Google

requisite to raise a weight -w; to a heigiit h, is Tepresemed hy
wh; this then is the number of units of work accumulated
in ttie body at the instant of projection. But since h is the
height through which the body must fall to ae(^uire the velo-
city «, therefore ti'=2^A (Art. 4T.); therefore h=^— ; whence

it follows that if U represent the number of units of work
aceumulated,

.(44),

Moreover it appears by the last article that this expression
represents the work accumulated in a body weighing w

Eounda, and moving with a velocity of ii feet, whaiev&r may
ave been the circumstances under which that velocity was
accumulated.

The product 1—)''° is called the vis viva of the body, so

that the accumulated work is represented by hah the vis

viva, the quotient (— J is called the mass of the body.*

67. To estmiaU the work aceumulated in a "body, or lost ly
it, as it passes from one velooify to another.

In a body whose weight is w, and which moves with a
velocity v there is accumulated a number of units of work

represented (Art. 66.) by the formula J— u'. After it has
passed from this velocity to another V, there will be accumu-
lated in it a number of units of work, represented by i— Y',

so that if its last velocity he greater than the first, mere
will have been added to the work accumulated in it a num-
ber of units represented by i— V"— ih-v''; or if the second

velocity be less than the first, there will have been taken
from the work accumulated in it a number of units repre-
sented by ^— y"— |— Y°. So that generally if IT represent

the work accumulated or lost by the body, in passing from
the velocity v to the velocity V, then

* Note (p) Ed. App.

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where the ± sign is to be taken according as the motion ia
accelerated or retarded.

68. 2'/ie work acffimiulated m a hody, whose motion is accele-
rated through am/ given space ly given forces ia equal to
the work whioh it would te neeesactry to do upon the hody
to cauae it to move iach again through, the same space
whm acted upon iy the same forces.

For it 19 evident that if with the velocity which a body
has acquired through any space AB by the
action of any forces whose direction ia from A
towards B, it be projected back again from B
towards A, then as it returns through each
■ successive small part or element of its path, it
will be retarded by precisely the same forces as those by
which it was accelerated when it hefore pa^ed through it ;
so that it will, in returning through each such element, lose
the same portion of its velocity as hefore it gained there ;
and when at length it has travei-sed the whole distance BA,
and reached the point A, it will have lost between B and A
a velocity, and therefore an amount of work {Art. 67.),
precisely equal .to that which before it gained between A
and B, Wow tiie work lost between B and A is the work
necessary to overcome the resistances opposed to the motion
thi'ough BA. The work accmnulated from A to B is there-
fore equal to the work which would be necessary to over-
come the resistances between B and A, or which would be
necessary to move the body from a state of rest, and with a
uniform motion, in opposition to these resistances, through
BA. Let tliis work be represented by TJ ; also let ti be the
velocity with which the body started from A, and V that

which it has acquired at B. Then will \ — (V—v^) repre-
sent the work accumulated between A and B,

.■.i|(Y--.-)=II, :.Y'-.-J§.

If the body, instead of being accelerated, , had been
retarded, then the work lost being that expended in over-
coming the retarding forces, is evidently that i

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THE AOCirainLATIOH OB WORK.

move tlie body unifonnly in opposition to these retarding
forces through AB ; ao that if this force be represented by

U, then, since i — {v' — V) ig in this case the wori: lost, we

ff

where the sign ± is to be talcen according as the motion i
accelerated or retarded.

69. The worJn acewmndaied m a l>ody which has moved
through any space acted wpon iy amy force, is emal to the
excess of the worh which has been dene i/mon tt iy thme
forces which tend to accderaie its motkm above that which
has ieen. done ttpon it l>y those which tend to reta/ed its

For let R be the single force wliich would at any point P
(see last fig.) be necessaiT tn move the body back asain
through an exceeding small element ot the same path (the
other forces impressed upon it remaining as before) ; then it
follows by Art. 54. that the work ol E over this element of
the path is equal to the excess of the work over that
element of the forces which are impresbed upon the body in
the direction of its motion abo\e the work of those
impressed in the opposite direction. Now this is true at
me!ry point of the path ; therefore the whole work of the
force R necessary to move the body back again from B to A
is equal to the excess of the work done upon it, by the
impressed forces in the direction of its motion, above the
work done upon it by them in a direction opposed to its
motion ; whence also it follows, by the last proposition, that
the ac^mwlatcd work is equal' to this excess. There-
fore, &c.

*70. If P represent the force in the direction of the
motion which at a given distance 8, measured along the

Eath, acts to accelerate the motion of the body, this force
eing understood not to be counteracted by any other, or tc
be the surplus force in the direction of the motion over and

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above any resistance opposed to it, then will / FdS be the
work whieli must be done in an opposite direction to ofer-
corae this force through the space S, or XJ=- j Pt^S,

2gfYd^
,\ by equation (40), Y'—v'=±—^ . . (47),

11. If the foi'ce P tends at first towards the direction in
which the body moves, so as to oGcelerate the motion, and
if after a certain space has been described it changes ita
direction bo as to retard the motion, and TJ, represent the
value of TT in respect to the former motion, and Ti the
velocity acquired when that motion has terminated, whilst
U, is the value of IT in respect to the second or retarded
motion, and if v be the initial and Y the ultimate or actual
velocity, then

'.— »- yf .

...^-..=?ffic:5)...,,,(48).

As TJj incj-eases, the actual velocity V of the body eon-
tinwally dimimahes ; and when at length TJ,=TJ„ that is
when the whole work done (above the resistances) in a
direction opposite to the motion, comes to equal that done,
before, in the direction of the motion, then V=w, or the
velocity of the body returns again to that which it had
when the force P began to act upon it. This is that gene-
ral case of reciprocating motion which is so freqnently pre-
sented in the combinations of machinery, and of which the
crank motion is a remarkable example,

*72. If the force which accelerates the body's motion act
always towards the same centre S, and S6 be taken equal to

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^ SB, it haa been shown (Art 55.) tliat ihe work

/I necessary to move the body along the curve from

/ 1 B to A, ia equal to that which would be necessary

,/---'"i to move it tlirough the straight line 5A. The

\ acGumulated work is therefore equal to that nece^-

\ i sary to move the body thi-ou^ the difference SA

\l of the two distances SA and SB (Art. 68,). If these

distances be represented by R, and S^, and P

represent the pressure with which the body's motion along

5A would be resisted at any distance It from the point S,

then/ ViISl will, represent this work. Moreover the woi'k

aGcv/muJ.ated in tlie body between A and B is represented
by ^— (V- — if), if V represent the velocity at B and v that
at A,

MR,

■ m-

73. Tlie work accumulated in the body while it descends
the curve AB, ie the same as that which it would acquire in
falling directly towards S through the distance Ai, tor both
of tliese are equal to the work which would be necessary to
raise the body from i to A. Since then the work accumu-
lated by the body through AB is equal to that which it would
accumulate if it fell tnrough AS, it foUows that velocity
acquired by it in falling, from rest, through AB is equal to
that which it would acquire in falling through A3. For if
T represent the velocity acquired in the one case, and V,
that in the other, then the accmnulated work in the first ease

"W" "W"

is represented by ^ — V,andthatintheBecondcaBebyi— Y,',

therefore i— V^ = i — Y,', therefore Y=V,.

From this it follows, that if a body descend, being pro-
jected obliquely into free space, or sliding from rest upon
any curved surface or inclined plane, and be acted upon only
by the force of gravity (that is, subject to no friction or
resistance of the air or other retarding cause), then the velo-

,y Google

city acquired by it in its descent is precisely the same as
though it had fallen vertioally through the same height.

74. Definition. The AfTGnLiE yelcicity of a hody which
rotates about a fixed axis is the arc which every particle of
the body situated at a distance unity from the axis deBcrihes
ill a second of time, if the body revolves imifonrAy ; or, if
the body moves with a vwruMe motion, it is the arc which it
would describe in a second of time if {from the Instant when
ita angular velocity is measured) its revolution were to
become uniform.

75. The accohilation of "woek is a body wincH

EOTATES about A FIXEO AXIS.

Propositions 68 and 69 apply to every case of the motion
of a heavy body. In every such case the work accumulated
or lost by the action of any moving force or pressure, whilst
the body passes from any one position to another, is equal
to the work whieii must be done in an opposite direction, to
cause it to pass back from the second position into the first.
Let us suppose TJ to represent this work in respect to a body
of any given dimensions, which has rotated about a fixed
axis from one given position into another, by tlie action of
given forces.

Let ct be taken to represent the anghlak velocht of the
body after it has passed from one of these positions into
another. Then since « is the actual velocity of a particle at
distance unity from the axis, therefore the velocity of a par-
ticle at any other distance p, from the axis ia «p,. Let f*
represent the weight of each unit of the volume of liie body,
and wi, the volume of any particle whose distance from the
axis is pi, then will the weight of that particle be (J-OTi ; also
its velocity has been shown to be ap„ therefore the amotint
of work accumulated in that particle is repr^ented by

t^-^Po or by t« -^.p, .

Similarly the different amounts of work accumulated in
the otlier particles or elements of the body whose distances
from the axis are represented by pj, p^, , . . and their

volumes by m-,, m„ m, . . . ., are represented by ia'-m^p,',
^cc'- s/fcjp/, &e. ; HO that the whole work accumulated is repre-

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ASGULAK VELOCITT.

, or ljyiB'^{m.p/+m,p,' + m,p/+ \.

The sum ''injf^-'rin.i^' + fn',^'+ . . . ., or 2mp" tabeu in
respect to all tho particles or elements which compose the
boQ;;^, is called its moment of iNEsiii in respect to the
particular axia aboiit which the rotation takes place. Let it

lie represented ty I ; then will ^a' . | — I . I, represent the

■whole amount of work accumulated in the tody whilst it has
been made to acquire the angular velocity « aom rest. If
therefore U represent the work which must be done in an
opposite direction to cause the hody to pass back from its
last position into its first,

I=U,

u

. (50).

If instead of the body's first position being one of rest, it
liad in its first position been moving with an angular velocity
«! which had passed, in its second position, into a velocity
a ; and if U represent, as before, the work which must be
done in an opposite direction, to bring this body back from

its second into its first position, then is ^' (-1 1~i"i\;;) I>
I, the work accumulated between the first

"*(?)('

and second positions ; therefore

*(7)(-

».■)!= ±U,

where the sign ± is to be taken according as ihe motion is
accelerated or retarded between the first and second posi-
tions, since in the one case the angular velocity increases
during the motion, so that a' is gi-eater than a,", whilst in the
latter case it diminishes, so that a° is less than o.^.

T6. If during one part of tlie motion, the work of the

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impressed forces tends to accelerate, and during another to
retard it, and the work in the former case be represented by
Ui, and in the latter by U,, then

=",'+

TVom this equation it follows that when TJ,=TJ„ or when
the work U, done by the forces which tend to resist the
motion at length, equals that done by the forces which tend
to accelerate the motionj then a=a^, or the revolving body
then returns again to the angular velocity from which it set
out. "Whilst, if TJj never becomes equal to TJ, in the course
of a revolution, then the angular velocity a does not return
to its original value, but is increased at each revolution ;
and on the other hand, if U, becomes at each revolution
greater than U„ then the angular velocity is at each revolu-
tion diminished.

The greater the moment of inertia I of the revolving
maes, and the greater the weight fj- of its unit of volume
(that IB, the heavier the material of which it is formed), the
less is the variation produced in the angular velocity a. by
any given variation of TJ or Fi— TT, at dift'erent periods of
the sijtme revolution, or from revolution to revolution ; that
is, the more steady is the motion produced by any variable
action of the impelling force. It is on this principle that
the fly-wheel ie used to equalize the motion of machinery
under a variable operation of the moving power, or of the
resistance. It is simply a contrivance for increasing the
moment of inertia of the revolving mass, and thereby
giving steadiness to its revolution, under the operation of
variable impelling forces, on the principles stated above.
This great moment of inertia is given to the fly-wheel, by
collecting the greater part of its material on the rim, or
about the circumference of the wheel, so tliat the distance
p of each particle which composes it, from the axis about
which it revolves, may be the greatest possible, and thus
the sum Zmp', or I, may be the greatest possible. At the
same time the greatest value is given to the quantity [j-, by
constructing the wheel of the heaviest material applicable
to the purpose.

What has here been said will best be understood in its
application to the crank.

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11. If we conceive a constant pressure Q to act upon tlia
E arm CB of tlie crank

in the direction AB of
the crank rod, and a
i^ constant resistance E
to be opposed .to the
revolution of the axis
0 always at the same perpendicular distance from that axis,
it is evident that since the pei-pendicular distance at which
Q acts from the axis is continually varying (being at one
time nothing, and at another equal to the mole length CB
of the arm of the crank), the effective pressure upon the
arm CB must at certain periods of each revolution exceed the
constant resistance opposed to the motion of that arm, and
at other periods fail short of it ; so that the resultant of
this pressure and this resistance, or the unbalanced pressure
P upon the arm, must at one period of each revolution have
its direction in the direction of the motion, and at another
time 02>posite to it. Representing the work done upon the
arm in the one case by U^, and in the other by TJ,, it follows
that if U,=Uj the arm will return in the course of each
revolution, from fiie velocity which it had when the work
TJ, began to be done, to that velocity again when the work
TJ, is completed. If on the contrary U, exceed U„ then the
velocih- will increase at each revolution ; and if TJ, be less
than U„ it wiU diminish. It is evident from equation (52),
that the greater the moment of inertia I of the body put
in motion, and the gi-eater the weight f- of its unit of
volume, the less is the variation in the value of a, produced
by any given variation in the value of TJ, — U, ; the le^
therefore is the variation in the rotation of the arm of the
crank, and of the machine to which it gives motion, pro-
duced by the varying action of the forces impressed upon it.
Kow the fly-wheel being fixed upon the same axis with the
crank ann, and revolving with it, adds its own moment of
inertia to that of the rest of the revolving mass, thereby
increasing gi'eatly the value of I, and therefore, on the prin-
ciples stated above, equalizing the motion, whilst it does not
otherwise increase the reeistance to be overcome, than by
the friction of its axis, and the resistance which the an
s to its revolution.*

* We shall hereafter treat fully of the orank and fly-wbeel.

, Google

78. Th& rotaUon of a hody cAout a Jixed mm wTten acted
■(ipon hy no oth&r inxyovng force thwn its weigM.

Let U represent the work neceaaaiy to raise it from its
second position into the first if it be descending, or from its
first into its second position if it be ascendmg^ and let a '^ -
its angalar velocity in the fii^t position, and a in *'"' '■"■'"
then by ec[uation (51),

I^ow it has been shown (Art. 60.), that the work necessai^
to raise the body fi-om its second position into the first if it
be descending, or from its first into its second if it be
ascending (its weight being the only force to be overcome),
is the same as wonld be necessary to mse its whole weight
collected in its centre of gravity from the one position into
tlie other position of its centre of gravity. Let OA repre-
sent the one, and OA, the other position of
the body, and G and G, the two correspond-
ing positions of the centre of gravity, then
' will the work necessary to raise the body
from its petition OA to its position CA„ be
eqnal to that which is nec^sary to raise its
wnole weight W, enpposed collected in G,
from that point to G, ; which by Article 56, is the same as
that necessary to raise it thron&a the vertical height GM.

Let now CG=CG,=A, let (9) be a vertical line through
0, let G,CD=^ and GCD=^ in the case in which the
bochr descends, and conversely when it ascends; therefore
GM=NN,=CN— CN,=A COS. fl— A cos. ^ when the body
descends, or =A cos. ^, — h cos. ^ when it ascends from the
petition AC to A0„ since in this last case GCD=fl, and
G,OD=S. Therefore GJI= ±h (cos. fl— cos. a,), the sign ±
being taken according as the body ascends or descends.
Xow U=W . ^K=±^h (cos. fl— cos. e.),

.■, by equation (51) !i^:=.a^-\-{ — ^^ j (cos. fl — cos. i^.

If M represent the volume of the revolving body M.i>-=W,

.■..■=.,-+fj^)(e„..-_COB...) (oB).

"When the body has descended into tlie vertical position,

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'0 MOMENT OF IMEBTIi

=0, 80 that (cos. a — coa. i),)=l— eos. a
t h.a.R ascmded iato that position fl=*,
;)=— (l+coe. K)=—2 coa.'i^.
In tlie first case, therefore,

,=2 Bin.'!',.
60 that (cob.

When
a — cos,

..=,.+(«?),„,j.....

. . (64).

In the second case,

..=<-(^)».-i.,..

. . . (55).

"When the body has descended or ascended into the hori-
zontal position fl=^, so that (cos. i — cos. i,)= — coe. ^,. But
it is to be observed, that if the body have descended into
the horizontal position, S, must have been greater tlian „,

and therefore cos. ^ mnst be negative and eqnal to — cos.
BOG, ; so that if we suppose ^i to be measured from CB or
OD, according as the body descends or ascends, then (cos.
6 — COS. ^,)=±co8. ^1, and we have for this case of descent
or aecent to a horizontal position

,.„.±?E™eoB.« (56.)

If the body descend from a state of i-est, a,^0.

.-. by equation (53) ii'=-^^(cos. S— cos. 6^) . . . (57).

Thus the angular velocity acquired from rest is, less as the
moment of inertia I is gi-eater as compared with the volume
M, or as the mass of the body is collected farther &om its
azis.

The Moment of Inertia.

79. Samng gwen the moment of meftia of a hody, or system
of lodges, ahout an aim fossi/ng through its centre of
gravity, to find its moment of vnertia fd>out cm axis, par-
add to the first, passing through amy other point im, the
hody or system.

Let m, be any element of the liody or system, i?i,AG a

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MOMEKT OF IKEIITIA.

71

M] plane perpendicular to tlie axis, aboui

/\s wMeh the moments are to be meaeured, A

/ x'x,^^ the point where this plane is intersected

^v;.-;__ „„..\;^ by that axis, and G the point where it is

intersected by the parallel axis passing
throngh the centre of gravity of the body. Join AG,
Am,, Gm„ and di-aw m,M, perpendicular to AG, Let

Aw,=|>„ Gm,=r„ GM,=a!„ AG=A.

K"ow (Enclid, 2—12.), Am.' = AG'+Gm,''+2AG . GM„

or f'=::h'+r'+2hx^.
if therefore the volume of the element be represented by
m.;, and both sides of the above ecLnation be mnltiplied by it,

And if OT,„ m,, m„ &e. represent the volumes of any other
elements, and p„ r„ x^ ; p„ r„ x„ &c. be similarly taken in
respect to those elements, then,

f'm,—h^m, + r'm, +^ke,m,,

p,^m, = h'm, + r'm, + 2/iai^m^,

&c.=&c.

Adding these equations we have, p*m, + i>,'m, + ?im,+ . . .

^A" {m^+m^+ m, + . . . . )-\-(^'m,+r^m^+r'm,,+ ....) +

2A(3!,»i,+«,m,+iB,m.j+ . . . ),

or sp°wi=A'Sm+Sr'OT+2A2am.

Now 2iCT?i is the snm of the moments of aU the elements

of the body about a plane perpendicular to AG, and passing

throngh the cenire of gravity G of the body. Therefore

'{Ai-t. Yl.) 2am=0,

.•. Sp'TO=A°2OT+2r'm,
Also 2p')ji is the moment of inertia of the body about the
given axis passing through A, and Sr'm is the moment of
inertia about an axis parallel to this, passing throngh the
centre of gravity of the body. Let the former moment be
represented by \ ; and the latter by I ; and let the volume
of the body s»i be represented by M,

;. I,=/t"M+I (58).

From which relation tlie moment of inertia (I,) about any
axis may be found, that (I) about an axis parallel to it, and
passing through the centre of gravity of the body being

80. The babius of oyeation. If we suppose \ to be the
distance from the axis passing through A, at which distance,

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Ti MOMEBT OF INERTIA.

if tlie whole mass of the body were colleetecl^ tlie moment of
inertia would remain the same, so that h'iA.^l^, then h^ ia
called the radiijs of oykatioh, in respect to that axis.

If ifc be the radius of gyration, similarly taken in respect
to the axis passing through G„ so that ^M=I, ,then, enbsti
tuting in the preceding eq^uation, and dividing by M,
h^^h' + h' (59).

Tlie following are examples of the determination of the
moments of inei'tia of bodies of some of tbe more common

.1 forms, about the axes passing through their cen-

s of gi'avity : tbey may thenee be found about any other
axes pai'allel to these, by equation (58).

*81. The moment (^ meHia of a tMn wrhiform rod alxmt t
ams perpendicular to its l&nffth and pasmig tl '

Let m represent an element of the rod contained between

two plane sections perpendicular to its

CL J-j-Tn::£;^;rfa faces, the area of each of which ia k, and

■"I ?t ■ n ■whose distance from one another is Ap,
I and let k and Ap be so small that eveiy

point in this element may be considered to be at the same
distance p from the axis A, about which the rod revolves.
Then is the volume of the element represented by KAp, and
its moment of inertia about A by Kp'Ap. So that the whole
moment of inertia I of the bai- is represented by Swp'Ap, or,
since « is the same throughout (the bar being unifonn), by
B2p'Ap ; or since Ap is infinitely small, it is represented by

the definite integi"al s / p'(7p, where I is the whole length

of the bar,

.■.I-«IKiQ-*(-i^1
orI=Ty;' (60).

*82. The moment of mertia of a thin rect<mgrdar lamma
ahout atn, ams, passing through its centre of' gravity, and
pwraUel to one of its sides.

It is evident that such a lamina may be conceived to be

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MOMEHT OF IHEKTIA.

73

made up of an infinite number of slendei
_ rectangular rode of equal length, each of
] wMeh will he bisected by the axis AB,
' and that the moment of inertia of the
whole lamina is equal to the sum of the
moments of inertia of these rods. Now if k he the section
of any rod, and I the length of the lamina, then the moment
of inertia of that rod is, by the last proposition, represented
by -^icl' ; BO that if the section of each i-od be the same, and
they be n, in number, then the whole moment of inertia of
the lamina is -^^w^. Now nic is the area of the transverse
section of the lamina, which may be represented by K, so
that the moment of inertia of the lamina about tlic axis AB
ited by the formida

I=AI"' (61).

'(W parallelvpipe-
centre of fframiy,

^83. T?ie moment of inmiia of
Aon about an ams, passing th
tmdpa/ralhl to either of Us edgi

Let CD be a rectangular paraUelopipedon, and AB an
axis passing through its centi-e of gi'avity and
irallel to either of its edges ;. also let ab be
I axis parallel to the first, passing through
the centre of m-avity of a lamina contained
by planes parallel to either of the faces of the
parallelopiped. Let a, h, o, represent the
three edges ED, EF, EG, of the parallelo-
piped, then will the moment of inertia of the lamina about
the axis a5 be represent-ed by i^b", where K is the trana-
verse section of the lamina (equation 61). Now let the
perpendicular distance between the two axes AB and oS be
represented by x. Then (by equation 68) the moment of
inertia of the lamina about tire axis AB is represented by
the formula '•>?l£-\--}^Eff, where M repr^ents the volume of
the lamina. Let the thickness of the lamina be represented
by A(B ; ;, M = oJAic, K = a^x ; ,•, m' in» of lam* = ahs?^x +
■^ah'i^x ; ;, whole m' in" of pai-allelopiped = a^sar'Aa) -|-
f'jiiJ'SAic ; or taking ^x infinitely small, and representing the
moment of inertia of tlie parallelopiped by I.

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74 MOMEHT OF IKEBTIA.

.■.I=T'^aSc(5°+c') (62).

*84. T^ vnommA of merUa of an upright lyricmgular prisr, i
about a vertical axis passing through its cmtre of gravity.

Let AB bo a vertical axis passing through tlie centre of
gravity of a prism, whose horizontal section is
an isosceles triangle having the equal sides ED
andEF.
Let two planes be drawn parallel to the face
Ij J DF of the prism, and containing between them
^ a thin lamma vq of its volume. Let Cm, the
perpendicular iBstance of an axis passing through
the centre of gravity of this lamina from the
axis AB, be represented by sc ; also let A(b represent the
thickness of the lamina.

Let DF= a, DG = 5, and let the perpendiculm- from the
vertex E to the base DF of the triangle DEF be represented
by c,

.■.EC = fc, Em=|c,-iK; a]so-gJ = — ,
.•.^^=z:-{|fl— k); also transverse section K of lamma = 5aiB,
.*. volume M of lamina = - i^^x)^x. Tlierefore by equa-
tions (58) and (61),

m' in" of km* about KS,= ^(^c—^Si.K^-}-^-^{^o~x-ft:^w\
.-.m' in' of prism about

oh y.+|c ^*' /.+I0

AB=— /(fo— iK)iK'&;+TJj— l{\o-<^dse.

Performing the integrations here indicated, and represent-
ing the inertia of the prism about AE by I, we have

l=r'^ aha iia'-^^ic') (63).

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MOMENT OF

*S5. TKa mom&n.t of inertia of a solid cylindsr ahout its
aids of-

CM'

Let AB te the axis of sucli a cylinder, whose radius AO
is represented by a, and its height by 5. Con-
ceive the eyhnder to be made «p of cylindrical
rings having the same axis ; let AP= p be the
internal radius of one of these, and let its thiclr-
_ ness PQ be represented by ^p, so thatp+Apig
Wja the exteral radius AQ of the ring. Tlien will
the volume of the ring be represented by
iri{p+ApY — wSp', or by ■Tor2pAp+(Apy] ; or if Ap
be taken exceedingly small, eo that (Ap) may vanish as com-
pared with 2pAp, then is the volume of the ring represented
by 2*SpAp,

Now this being the case, the ring may be considered as an
element AM of the volume of tlie sohd, every part of which
element is at the same distance p from the axis AB, so that
the whole moment of inertia 2p°AiI of the cylinder =
2p'(2irJpAp) =;VS2p'Ap,

:.l=2*hJ'p'df>=^M {64:).

*86. The moment of inertia of a hollow cyli/nderr about its
axis of s

Let «i be the external radius AO, and «, the internal
radius AP, and 6 the height of the cylinder ;
then by the last proposition the moment of in-
ertia of the cylinder CD, if it were solid, would
be ^ia'; also the moment of inertia of the
cylinder Pli, which is taken from this solid to
form the hollow cylinder, would be -^wSo-j'. Now
let I represent the moment of inertia of the hol-
low cylinder CP, therefore l-^-^ia^^^ha^,

:.\=^l{a^ — a^)=^l{a^—a'){a^+a^)==.^}i{a^ — «,)

{<..+<..)(<.,■+«,■).

Let the thickness a, — a, of the hollow cylinder be repre-
sented bv <?, and its mean radius \{fl^-'ra^) by K, therefore

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i of a ffj/Under aiout an axie
of grm^ity, and p&r^&ndicular

Substituting these values in tlie preceding equatio
tain

I=2*5(!RjE'+ic'[ (65).

■^87. The mommt of

passing tkrough its centre
to its eaaia of symmetry..

Let AB be such an axis, and let PQ represent a lamina

^ contained between planes perpendicular to

-f^^it n~^ tliie axis, and exceedaugly near to each other.
ii' I ^ Let CD, tiie axis of the cylinder, be repre-
■i~l-'"-i— ■ — I — sented by 5, its radios by a, and let Cil=a!.
Take ^x to represent the thietneBs of the
lamina, and let MP^?/. !Now this lamina
may be considered a rectangular parallelo
piped traversed through its centre of gravity by the axis AB ;
therefore by equation (62)it8 moment of inertia about that axis
is represented by ^{'^i'>)b{^y)\'b^ + {^yy}=ii\b''>/+4^'\^x.
Now the whole moment of inertia I of the cylinder abouE
AB is evidently equal to the sum of the moments of inertia
of all such lammte ■

Also, since x and y are the co-ordinates of a point in a
circle from its centre, therefore j/= («'— a)'*)*. Substituting
this value of y, and integi'ating according to the well known
lilies of the integral calculus,* we have

l=^ha\a'+W) (66).

*88. The moment tfmeriia of a cone ahout its axis of

The cone may be supposed to be made up of lamina, such
as PQ, contained by planes perpendicular to
the axis of symmetry AB, and each having its
centre of gravity in that axis. Let BP=ic, and
let AiB represent the thickness of the lamina,
and y its radius PH. Then, since it may be
considered a cylinder of very email height, its
moment of inertia about AB (equation 64) is
represented by ^y^^x. Now the moment of
urch'e Di£f. and iHtcg. Caluuhis, Arts. 143, 149.

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OF mSBTIA. I i

inertia I of tlio whole cone is equal to the aura of the mo-
menta of all such elements,

Let the radius of the base of the cone be represented bj

X h

re-=t-, 1
ij a a

.■.l=,\^M (67).

89. Tke moment of inertia of a t^here about one of its

Let C bo the eenti-e of the sphere and AB the diameter
about which its moment is to be determined.
Let PQ be any lamina contained by planes
perpendicidar to AB ; let CM=iB, and let Aai
~r represent the thickness of the lamina, and y ita
radius ; also let CA=ra ; then since this lamina,
being exceedingly thin, may be considered a
cylinder, its moment of inertia about the axis AB is (equa-
tion 64) ^j/'Asi ; and the moment of inertia I of the whole
sphere is the sum of the moments of all such laminse,

Kow by the equation to the circle 'f^^a'—x', therefore
y*=ts'— 2a.V+a?'. If this value be substitiited for y', and
the integration be completed according to the common
methods, we shall obtain the equation,

I=A"' (68).

to. The moment of inertia of a corie about an a
through its (Kmtre of gra/iyhPy a/nd jper^endimlar to its
of syw/meiyry.

Let CD be an axis passing through the centre of gra^nty

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fG of the cone, and perpendiculai" to its axis of
symmetry, and let GP the distance of the lamina
j from G, measured along the axis, he represented
hy X ; also let the thickness of the lamina be re-
presented hy AiB, iJow this lamina may be con-
sidered a cylinder of exceedingly smail thick-
' ness. If its radius be represented by y, its mo-

ment of inertia about an axis parallel to CD paasmg through
its centre, is therefore (equation 66) represented by
'b'y'lv'+K'^fi'^j orifiia; be a^ilmed exceedingly small,
it IS represented by ^y'^3}. INow this being the moment of
the lamina about an axis parallel to CD, passing throng its
centre of gravity, and the distance of this axis from CD be-
ing X, and also the volume of the lamina being i^Aaf, it fol-
lows (equation 58), that the moment of the lamina about CD
is represented by iryVia7+i*^''ia;=<)j'V+iyj'iic.

Now the moment I of 3ie whole cone about CD equals
the sum of the moments of all such elements,

.•.I=*2(^V + i/)AiB.

Now if OS be the radius of the base of the cone and i its
height, then since BG=|5,

% — X i 5, a S3, 5,

:.- = -: .•.a!=-(f«— y) and Aic= Aw;

y a' a" '' a ■

.■.1=.-Jtf' j ^,(|«-2/)y +i,/ 1 */,

91. The moment of mertia of a segment of a sphere about
a dia/meter pa/raUel to the plane of section.

Let ADBE represent any such portion of a sphere, and
_i, AB a diameter parallel to the plane of section.

/^^ZCn Let CD=(t, OE=J, and let PQ be any lamina

P^^^^~"^_ contained by planes parallel to the plane of
ST yx.TSj* section : let the distance of the lamina from

Vr'_f-->^ C=iB, and let its thickness be As; and its radios
y. Then considering it a cylinder of exceedhig small thick-

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MOVIKG F0E0E8. 70

ness, its moment of inertia about an axia passing through its
centre of gravity and parallel to AB, is represented (equa-
tion 6Q) by i^y'lv'+it'^xy'li^a!, or (neglecting powers of Aai
above the first by ^y*^ie. Hence, therefore, the moment of
this lamina about the axia AE is represented (equation
68) by rry\i^)^ +^y^i^, or by * \y^x' +^*\ax ; now the
whole moment I of inertia of ADBE about AB is evidently
equal to the sum of the moments of all such laminfe,

:.l=r2

If'^" +^/\^''e= ^(Jv + h/)^-

Now y'=:a'—af, therefore yV+^°=i|2»V— 3a)' + a'5 .
Substituting this value in the integral and integrating, we
have

I=Tf'5*S16(x'+15a'J+10a'6'~96'J* (70)

THE ACOELEEATION OF MOTION BY GIVEN
MOVING EOECES.

93. If the forces applied to a moving body in the direc-
tion of its motion exceed those applied to it in the oppoaite
direction (both sets of forces being resolved in the direction
of a tangent to its path), the motion of the body will be ac-
cel&ratm; if they fall short of those applied in the opposite
direction, the motion will be retarded. In either case the
excess of the one set of forces above the other is called the
MOVING FOHCB upou the body : it is measured by that single
pressure which being applied to the body in & direction op-
posite to the greater force, would just balance it ; or which,
tad it been applied to the body (together with the other
forces impre^ed upon it) when in a state of rest, would have
nuuntained it in that state ; and which, therefore, if applied'
when its motion had commenced, would have caused it to
pass from a state of vmiabU to one of vmform motion. Thus
the moving force upon a body which descends freely by gra-
vity, is measured bv its wdght, that is, by the single force
which, being applied to the body before its motion had com-
menced in a direction opposite to gravity, would just have
supported it, and which being apphed to it at any mstant of

• Note {q) Ed. App.

, Google

so EELATIONB OF

its descent, would have caused ita motion at that instant to
pass from a state of variable to a state of uniform motion.
If the resistance of the air upon its descent he talsen into
account, then the moving force upon the body at any instant
is measured by that sin^e pressure which, being applied up-
wards, would, together with the Tceistance of the air at that
instant, just balance the weight of the body.

A moving force being thus understood to be measured by
9t presatere,* being in fact the -imbakmoed pressure upon the
moving body, the following relations between the amount of
a moving force thus measured, and the degree of acceleration
produced by it will become intelligible. These are lams of
motion which have become known by experiment upon the
motions of the bodies immediately around us, and by obser-
vation upon those of the planets,

93. When the moving force upon a body remains con-
stantly the same in amount (as measured by the eCLuivalent
pressure) throughout the motion, or is a imiform moving
force, it communicates to it equal additions of veloci'^ in
equal successive intervals of time. Thus the moving force
upon a body descending freely by gravity (measured by its
weight) bemg constanUy the same in amount throughout its
descent (the resistance of the air being neglected), tlie body
receives from it equal additions of velocity in equal succes-
sive intervals of time, viz. 32^ feet in each successive second
of time (Art. 44.).

94. The increments of velocity communicated to eq-ual
lodiss by unequal movingforces (supposed wniform as above)
are to one another as the amounts of tliose moving forces
(measured by their equivalent pressures).

Thus let r and P, be any two unequal moving forces ujion
two equal bodies, and let tnem act in the directions in which
the bodies respectively move ; let them be the only forces
tending to communicate motion to those bodies, and remain
constantly the same in amount throughout the motion. Also
let f and /; represent the addition^ velocities which these
two forces respectively communicate to those two equal
hodi^ in each successive second of time ; then it is a law of
the motion of bodies, determined by observation and experi-
ment, that P : P, ::/ :/,.

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ASD MOTION".

81

If one of the moving forces, as for instance P„ "be the
•weight W of the body moved, then the value f, of the
increment of velocity per second corresponding to that
moving force is 32' {Art. 4i.) represented by g,

W

.■•p=t/ («)•

95. If the amount or magnitnde of the moving force does
not remain the same throughout tlie motion, or if it be a
variahle moving force, then the increments of velocity com-
municated by it in equal successive intervals of time are not
equal; they increase continually if the moving force
increases, and they diminish if it diminishes.

If two unequal moving forces, one or both of them, thus
va/riahle in magnitude, become the moving forces of two
equal bodies, the additional velocities which they would
communicate in the same interval of time to those bodies,
if at any period of the motion from vcmahU they become
wdform., are to one another (Art. 94.) as the respective
moving forces at that period of the motion.

Thus let/ and/i represent the additional velocities which
wtmld thus be commimicated to two equal bodies in one
second of time, if at any instant the pressures P and P„
which are at that instant the moving forces of those bodies,
were from wwiahle to become constant pressures, then
(Art. 94.),

P:P.::/:/.

This "being true of any two moving forces, is evidently true,
if one of tbem become a constant force. Let Pj represent
the weight W of tlie body, then will f, be represented

.•.P:W::/:j,.

Let the moving force P be supposed to remain constant
during a number of seconds or parts of a second, repre-
sented by M, and let aV be the increment of velocity in
the time ^t on this supposition. ISTow f represents the
increment of velocity in each second, and aV the increment
of velocity in M seconds : moreover the force P is supposed
constant c"iring M, so that the motion is unifo'n/ily accele-
rated during that time (Art. 44.).

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EELATI0N8 OF

.•./A«=AT, ;,/=--

Now tliie is true (if tlie supposition, that P remains constant
during the time M, on which it is t'oimded, he true), how-
ever small the time M may be. But if this time be
infinitely small, the supposition on which it is founded is in
all cases true, for P mayin all cases be considered to remain
the same during an iniinitely email period of time, although
it does not remain the same during any time which is not

iniinitely small. Now when a; is infinitely small — = -y- ;
generally therefore /"— -32 -

If V increase as the time i increases, or if the motion bo
accelerated, then ^ is necessarily a positive t[uantity. If,
on the contrary, V diminishes as the time increaseH, then
-t2 is negative ; so that, generally,

f--% m.

the sign ± being taken according as the motion is accele-
rated or retarded. Substituting this value of f in the last
proportion we have in the case, in which P represents a
variable pressure,

• (W).

The principles stated above constitute the fundamental r
tions of pressure and motion.

96. The velocity V at any instant of a body moving with
a va/riable motion, being the space which it would describe
in a second of time, if at that instant its motion were to
become uniform, it follows, that if we represent by ^t any
number of seconds or parts of a second, beginning irom that
instant, and by Ag, the space which the body woiud describe

* Note (r) Ed. App.

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in tlie time A(, if its motion continued uniform from the com-
mencement of that time, then,

YM=^S, .■.V=-;?.

' AS

Now this 18 true if the motion remain uniform during the
time M, however small that time may be, and therefore if it
be mfinitdy small. But if the time a; be mfinitdy email,
the motion does remain uniform duiing that time, however
variable may be the moving force ; also when M is infi-

„ AS <;s ^, „

nrtely email, Tj = ;^- Ihereiore, generally,

. (T4).

The equations (73) and (74) are the fundamental equations
of dynamics : they involve tbose dynamical resnlts which
have been discussed on other principles in the preceding
parts of this work,*

The Desceht of a Body tipon a Odbve.

*97. If tkemovingfor(xT i^on a lody varies dwectly mite
distance ai any PvmefTom a given point towwrds which it
faUs, tkmi the whole time of the tody's foMng to thai
point wiU ie the sa/ms, whateo^ may be the 3ista/>ioefrom
which itfaUs.

Let A be the point from which the body falls, and B a
point towards which it falls along the path
APB, which maybe either curved or straight;
also let the body be acted upon at each
point P of its path, by a force m the direc-
tion of its path at that point which varies as

le inverted, and multiplied by the former, ne

.■.¥'_„'=+% fvdS,
Tthicii 19 Identical with equation (41).

, Google

its distance BP, measured along the. patli from B ; the time
of falling to B will be tlie same, whatever may be tlie dis-
tance of the point A from wMch the body falls.

For let BI'=S, and let the force impelling the hody
towards E be represented by c8, where c is a constant quan-
tity ; suppose the body, instead of falling from A towards
B, to be projected with any velocity from B towards A, and
let.« be the velocity acquired at r, and V that at A, and
let BA=S„ then by equation (47),

^^v'=-UjcMS^-

Suppose now the velocity of projection from B to have
been such as would only just cany the body to A, so tJiat
V=:0,

.•.^'=t(S,'-S') (75).

"Now by equation (T4),

't/{|

dS

■■•'~'(f)*(S--^-)*;

and if JT represent the whole time in seconds occupied in
the ascent of the body from B to A,

It is clear that the time required for the body's descent
from A to B is equal to that necessary for the ascent from
B to A, so that the whole time required to complete the
ascent and descent is equal to T, and is represented by the
formula

-(!)■

• (W).

, Google

Now tbiB expression does not contain 8„ i. e. the distance
from wliich the body Mis to B ; the time T is the same
therefore, whatever mat distance may he.

The Simple PEsnnLTiM.

98. If a heam/ particle P he vmagimd to he suspended J)'
pMni Ql>ya thread without weight, amd aU<med to osc

freel/g^ hd so as to deviate hut UMe on either side of the
vertieal, then will its osolUaiions, so long as they a/re tkus
smaU, 06 performed in, the same time whaieoer their ampU-

Por let the inclination POB of OP to the vertical be repre-
sented by ^, and let the weight w of the particle
P, wMcn acta in the direction of the vertical VP,
a resolved into two others, one of which is in the
direction OP, and the other perpendicnlar to that
direction : the former will be wholly connteracted
by the tension of the thread OP, and the latter will
be represented by w sin. TPO=^c sin. fl ; and, act>
ing in the direction in which the particle P moves, this will
be the WJA<?fo impressed mowjw force upon it (Art. 92.) Now
80 long as the arc ^ is small, this arc does not differ sensibly
from its sine, so that for amall oscillations the impressed mov-
ing force npon P is represented by w6, or by— -y-, or by -=-,

if I represent the length OP of the suspending thread, and S
the length of the arc BP. INow in this expression w and I
are constant throughout the oscillation, the moving force va-
ries therefore as S. Hence by the last proposition, the email
OBciUations on either side of OB are isochronous, since so long
as they are thus small, the impressed moving force in the
direction of the motion varies as the length of the path BP
from the lowest point B. Since in the last proposition the
moving force was assumed equal to c8, and that here it is

represented by -j-S, therefore in this case c=— . Substitut-
ing this value in equation (76),

Mf' (")•

A single particle thus suspended by a thread without

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86 THE PAEALLELOGKAM OF MOTION.

weight, is that which is meant bj a simtle pendulcim. It i&
evident that the time of oscillation increases with the length
I of the pendulum.

iMPITLSrVE FOECE,

99. If any number of different moving forces be applied
to as many equal bodies, the velocities communicated to
them in the same exceedingly small interval of time, -will be
to one another as the moving forces. For let P„ P„ repre-
sent the moving forces, and /'„ f^, the additional velocities
they would communicate per second if each moving force
remained continually of the same magnitude (Art. 93.), then
would tf„ ^'„ be the whole velocities communicated on this
supposition in t seconds ; let these be represented by V„ Y, ;
therefore by Art. 94.

V,:V,::f,:f.::tf,:tf.::Y.:Y..

The proposition ie therefore true on the supposition tliat P,
and Pj remain constant daring the interval of time t ; but
if t be exceedingly small, then whatever the pressures P,
and P, may be, t£ey may be considered to remain the same
during that time. Therefore the proposition is true ^m^«^y,
when, as above, the moving forces are supposed to act on
equal bodies, or successively on the same body, through
equal exceedingly small intervals of time.

Moving forces thus acting through exceedingly small in-
tervals oftime only, are called iMPtrLsrvE foeces.

Tee Paealleloabam of Motion.

100. ^ two trnpuisive forces V,, Pj, wliose direetions are AB
i AO, he vtrupressed at the same time 'Wpon
"y ai A, whieh if made to. act upon it

^

___ I AC mthe same given time, then

will the hofhj le made, ly the s-kmdianeous action of these
impidsive forces, to desorihe i/n that time the diagi^ial AD
ojthe faralMogra/m,, of which AB amd AO ore a^OG&id
sides.

For the moving forces P^ and P, acting separately upoi:

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OF SIWULTANEOUi MonOINS. S.

the same body through equal infinitely small times, cone (nu-
nicate to it velocities ■which are (Art. 99.) as those foiceS;
therefore the spaces AB and AO described -with these velo-
cities in any given time are also as those forces. Since then
AB and AC are to one another as the pressures Pi and P,,
therefore by the principle (Art, 2.) of the parallelogram of
pressui'es, the resultamt E of P, and P, is in the direction of
the diagonal AD, and bears the same proportion to P, and
P, that AD does to AB and AC.

Tlierefore the velocity which the resultant K of Pj and P,
wotdd communicate to the body in any exceedingly small
time is to the velocities which JP, and P, would sepai'ately
communicate to it in the same time as AD to AB and AC
{Art. 99.), and therefore the spaces which the body would
describe uniformly with these three velocities in any equal
times are in the ratio of these three lines. But AB and AC
are the spaces actually described in the equal times by rea-
son of the impulses of P, and P,. Therefore AD is the space
described in that time by reason of the impulse of E, that is,
by reason of the simultaneous impulses of Pi and Pj.

101. The independence of

It is evident that if the body starting from A had been
made to describe ABin a given time, and then

y~^__-7° had been made in an equal time to describe
^;-"""^l^/ BD, it would have arrived precisely at the same
point D to which the simidtaneous motions
AC and AB nave brought it, so that the body is made to
move by these simultaneous motions precisely to the same
point to which it would hare been brought by those motions,
communicated to it successively, but in half the time. Tlie
following may be taken as an illustration of this principle of
the independence of simultaneous motions. Let a canal-boat
-n J be imaOTied to extend across the whole
V"" / ">\;;7 width of the canal, and let it be supposed

i-""-^- that a person standing on the one bank at

"*" ° A is desirous to pass to a point D on the
opposite bank, and that for this pui-pose, ae the boat passes
him, he steps into it, and waUis across it in the direction
AB, arriving at the point B in the boat precisely at the in-
stant when the motion of the boat has carried it through
BD ; it is cleai' tliat lie will be brought, by the joint effect

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88 THE POLYGON OF MOTION.

of hijS <non motion across the boat and the hoafs motion
along the canal, to the point D (having in reality described
the diagonal AD), which point he would have reached in
doable the time if he had walked across a bridge from A tc
B in the same time that it toot him to walk across the boat,
and had then in an cq^ual time walked from B to D along
the opposite side.

The Poltgon of Motion.

103, Let any number of impnlees be communicated simul-
taneously to a body at 0, one of which
would cause it to move from A to 0 in a
given time, another from B to 0 in the
same time, a third from C to O in that time,
and a fourth from D to 0. Complete the
parallelogram of which AO and BO are ad-
jacent aides ; then the impulses AO and BO would simulta-
neously cause the body to move from E to 0 through the
diagonal EO in the time spoken of. Complete the parallelo-
gram EOCr, and draw its diagonal OF, men would the im-
pulses EO and CO, acting Bunultaneoualy, cause the body to
move through FO in the given time : but the impulse EO
produces the same effect on the body as the impulses AO
and BO ; therefore the impulse AO, BO, and CO, will
together cause the body to move through FO in the given
time. In the same manner it may be shown that the im-
pulses AO, BO, 00, and DO, will together cause the body to
move through GO in a time equal to that occupied by the
body's motion through any one of these lines.

It will be observed that GD is the side which completes
the polygon OAEFG, whose other sides OA, AE, EF, FG,
are respectively equal and parallel to the directions OA, OB,
OC, and OD, of the simultaneous impulses.

Instead of the impulses AO, &c. taking place simultane-
ously, if they had been received successively, the body
moving firet from O to A in a given time ; then through
AE, which is equal and parallel to OB, in an equal time ;
then through EF, which is equal and parallel to OC, in that
time ; and lastly through FC, which is equal and parallel to
Oi), in that time, it would have arrived at the same point G,
to which these impulses have brought it simultaneously, but
after a period as many times greater as there are motions, so

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that the pi-inciple of the independence of sinmltaneoua
motions obtains, however great may "be the number of sneh
motions.

The PKraoiPLE of D'Alembekt.

103. Let W,, W,, W^, &c. represent the weights of any
number of bodies in motion, and P„ P,, Pj, &c. the moving
forces (Art. 92.) upon these bodies at any given instant oi
the motion, i. e. the unbalanced presanres, or the pressures
■which are wholly employed in producing their motion, and
pressures equal to ■which, applied in opposite directions,
■would bring them to rest, or to a state of uniform motion.

■WWW
Then (Art. 95.), P.= -^/, P,= --/„ P,= -^/„&c.

■where /|,/'j,/'„ &c. represent the additions of velocity ■which
the bodies would receive in each second of time, if the
moving force upon each were to become, at the inetant at
■which it is measured, an wnxform mo'ving force. Suppose
these bodies, whose weights are Wj, W„ W„ &c, to form a
m/stem of bodies united together ty any conceivable mecha-
nical connection, on whidi system are impressed, in any
way, certain forces, whence result the unbalanced pressures
P,, P„ Pj, &c. on the moving points of the system. Now
conceive that to these moving points of the sratem there are
applied pressures respectively equal to P„ P„ Pj, &c. but
each in a direction opposite to that in ■which the motion of
the corresponding point is accelerated or retarded. Then
■will the motion of each particular point evidentlypass into
a state of ■wdform motion, or of rest (Art. 92.). The whole
system of bodies being thus then in a state of uniform
motion, or of rest, the forces applied to its different elements
must be forces in equilibrium.

Whatever, therefore, were the forces originally impressed
upon tJie system, and causing its motion, they must, together
with the prrasures P,, P„ P„ &c. thus applied, produce a
state of equilibrium in the system ; so that these forces (ori-
ginally impressed npon the system, and known in Dynamics
as the IMPEE8SED forces) have to the forces P„ P„ P,, &c.,
when applied in directions opposite to Uie motions of their
several points of application, the relation of forces in equili-
brium. The forces P„ P„ P„ &c. are known in Dynamics
as the EFTEorrvE foeces. Thus in any system of hodies
mechanically connected in, any way, so that their mx)tions

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90 THE PEIHCIHE J)l' DaO;MBEET.

map TwutuaUy iiijkience one (mother, if forces equal to tfn
^&ctiv6 forces were a/ppUed in directions ojyposite to tlisi/r
actual mrections, these would he in eqmliori'um with the
impressed forces, which is the principle of ly Al&inhert,

lOi. The work accMirmlated in a movinff tody through amy
space is equal to the work which must he done upon it, in
<m opposite direction, to overcome the effeatvoe force upon
it through thai apace.

This 18 evident from. Arts. 68, and 69., since the effective
force is the unhalanced pressure upon the body.

If the -work of the effectire force be said to be done upon
tiie body,* then the work of the effective force iipon it is
equal to the work or power accumulated in it, and this work
of the effective force may be all said to be actually accu-
mulated ill the body as in a reservoir.

Motion of Translation.

DEFnjmoN. — When a body moves forward in space, with-
out at the same time revolving, so that all its parts move
with the same velocity and in parallel directions, it is said to
move .with a moidon of tramlation only.

105. In order that a hod/y mwy move with a motion of tra/ns-
lotion only, the resultant of the forces impressed upon it
must home its direction through the centre of gramiy of

For let w„ w„ w„ &c. represent the weights of the parts
or elements of the bi^dy, and let y represent the additional
velocity per second, which any element receives or would
receive if its motion were at any instant to become uniformly
accelerated. Since the motion is one of translation only,
the value of / is evidently the same in respect to every
other element. The effective forces P,, P,, P„ &c. on tlie
different elements of the body are therefore represented by

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MOTION OF KOTATION, 91

Now the forces Fj, Pj, P„ &e. are evidently parallel pres-
sures. Let X be the distance of the centre (see Art, 17.) of
these parallel pressures from any given plane ; and let x„ x^
x„ &c. be the perpendicular distances of the elements w,, w,,
«i„ &c, that is, ot the points of application of P„ P„ P,, &c.
from the same plane. Therefore (by eq^nation 18),

jP.+P,+P,+ ] X=Pa+Pa+Pa+ i

. -y_W,a;, + W;^, + Wa!K„+ . . ■ .
" ~ iOj+Wj+wij-l- ....

But this ia the expression (Art. 19.) for tlie distance of the
centre of gravity from the given plane ; and this being tme
of any plane, it follows that the cent/re of the parallel pres-
sures P„ P,, Pj, &c. which are the effectwe forces of^ the
system, coincides with tlie centre of gravity of the system,
and therefore that the resultant of the effective forces passes
through the centre of gravity. Now the resultant of the
effective pressures must coincide in direction with the result-
ant of the itnpressed pressures, since the effective pressures
when applied in an opposite direction are in equilihrvum
with the impressed pressures (by D'Alembert's principle).
Tlie resultant of the impressed pressures must therefore have
its' du-ection through the centre of gravity. Therefore, &c.

Motion of Rotation about a fixed Axis.

106. Let a rigid body or system be capable of motion
about the axis A. Let to,, OTj, to,, &c. represent the volumes
of elements of this body, and i* the weight of each unit
of volume. Also let/„/„y5, &c. represent the increments
of velocity per second, communicated to these elements
respectively by the action of the forces i/mpressed upon the
system. Let P„ P,, P„ &c. represent these impressed forces,
and »,, «„ &c. the perpendicular distances from the axis at
which they are respectively applied.

Now since nm„ fwi,, i^m,, &a. ai'e the weights of the ele-
ments, SindJ^^^f^, &a. the increments of velocity they receive

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MOTION OF 30TATI0N.

per second, it follows that ^—^ f,. it!!h f., — if,, &c. are

9 9 S

the effective forces npon them (Art. 103,). Let p„ p„ pj, &c.
represent the distances of these elements respectively from
the axis of revolution, then since their effective forces are
in du'ections perpendicular to these distances, the moments

of these effective forces ahont the axis are — ^/"iP,, -— ^ /"iPu

^/,P„ &c. Also P, »,, Y^„ P^,, &c. are the moments of

g
the impressed forces of the system ahont the axis. !Now the
impressed forces P„ P,, P„ &e., together with the resistance
of the axis, -wliich is indeed one oi the impressed forces, are
in equilibrium with the effective forces by D'Alembert's
principle. Taking then the axis as the point from which the
momenta are measui'ed, the sum of the moments of P„ P„
&e. must equal the sum of the moments of the effective
forces, or

«h/.p, + 'f^/,,.+ ....=P,y,+P,y,+ ..,.

Now let y represent that value of f„f„ &c. which coiTes-
ponds to a distance unity from the axis. Since the system
is rigid, and f, /■„ /„ &c. represent arcs described about
it in the same time at the dinerent distances 1, pi, p,, &c. it
follows that these arcs are as their distances, and therefore
thaty,=yp„y,=yp,„_/^,=ypi, i&c. Substituting these values
in the preceding equation, we have

5»,/p,- + em,/,,'+ =P>y,+Pj,,+ ;

.•./5Kp,-+«»,p,'+ ]=V,p,+V,p,+

S

or/-2mp'=2P^ , , ,,

.-./=? ^ (TO),

where I represents the moment of inertia of the mass about
its axie of revolution.*

* If o represent the angular Telocity, or the velocLtj of an element at St»
lance unity, then by equation (12),/= + J, .-. „^ = + ^SPpa;

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AG by G,

MOTION OF EOTATION.

forces P be tbe weights of the parts
of tbe body and fl be, in any position of
the body, tbe inclination to me vertical
A.y of the line AG, drawn from A to the
centre of gravity G, then since the sum of
the moments of the weights of the parts is
equal to the moment of the weight of the
whole mass collected in ita centre of
gi-avity (Art. IT.), we have, representing

sP^=]Mf* . GG,=]VI^i . G. sin. fl
MG

therefore (^equation ■T8),/=5f— -j^ sin. &

■ m-

108. To find the resultant of the efi'eotwe forces o
which T<

The resultant of the effective forces upon a body which
revolves about a iixed axis, is evidently equal to that single
force which would just be in equilibriimi with these if there
were no resistance of the axis. Let K be that single force,
then the moment of E. about any point must equal the sum
of the moments of the effective forces about that point.

Talie a point in the axis for the point
about which the moments are measured,
and let L be the pei-pendicnlar distance
from A of the resultant E. Now, as in
Art. 106. it appeal's that the sum of tlie
moments of the effective forces about A is

represented by f-^mf,

^''-tfl^fip'^'-

Now pa is the Yulocitj of a point at distaQi;e p, therefore Fpu is the v:tirJi
(Art. 60.) of the force P per seeoncl ; therefore / Fpadt is the worlt of P
{eqnation 40) in the time ?, which is reprpoentcd byU, therefore Oi' — Oj*
■^^t-^^vhieh eocresponda with the leralt alieadj obtained. See equation

, Google

M

.-. BL=/=J».," (80).

To deterraine the value of E let it be observed that tlw
effective force -/w-iP, on aay particle rtby, acting in a direc-
tion *»,m„ perpendicular to the distance Aw*, from the axia
A, may be resolved into two othera, parallel to the two
rectangular axes Ay and Asc, each of which is equal to the
product of this effective force, whose direction is )i,wi„ and
the cosine of the inclination of n^m^ to the corresponding
axis. !Now the inclination of m^n^ to Aic is the same as the
inclination of Am, to Aj/, since these two last lines are per-
pendicular to the two former. The cosine of this inclination

equals therefore— i or ^, if AN,=y,. Similarly the cosine
Am, p,

of theinchnationof w,wi, to Awequalfl !or — , if AM =^.

Am, p.
The resolved parts in the directions of A.x and Ay of the

effective force * /w^iP, are therefore - fm,?. ^, and _ fm,p,

9 9 h 9

~, or - /ffi w and - fmx

Similarly the resolved parts in the directions of AiB and

Ay of the effective force upon m, are -ffTtM, and - fin{G^,

9 9

and so of the rest.

The sums X and T of the resolved forces in the directions
of k.x and Ky respectively (Art. 11.) are therefore

^M3/.+^>,3/,+^>=y.+ .... -X,

and!:^/m,iC,-^!^/m,^, + *^/m,ie,+ . . . =T;
9 9 9

or^^/Smj^. + Tft^yi-t-m^j/.-f i=X,

9

and t^ / 5m,iB, + m^, + «iji», + }=:T.

Now let G, and G, represent the distances G^G and G,G
of the centre of gravity of the body from Ay and A* respec-
tively, and let the whole volume or the body be represented
by^,

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MOTION OF KOTATIOW. 95

.*. (equation 18), 'KOi,=m^y,-\-m^y^+'»i,y,-\- . . . .,
MG-,=m,iB,+m.,iC,+m,(C34- . . . .;

/. X=-/MG„ Y= VilG. (81)-

'(^t.u. * ^"^ ^^'^^ ^^O' li =^ VX' + Y^ therefore

i XjO Now if G be the distance AG of the

"i centre of gravity ftom A, G= VG'-\-Qr^,

.•.K=-/MG (82).

Substituting in ecLuation (82) the value of/ from equation
(78,) we have

11 = 1^ (88).

And substituting in equation (80) for R its value fVom
equation (82),

■■■'^=is («*).

where L ia the distance of the point of aj^plication of the
resultant of the effective forces from the sixia.

Now let & be the inclination of the resultant H to the
axis AiB,

;. (Art. 11.), R COS. '!=X, R sin. 6=Y,

T

.■.tan. ^=y ; but by equations (81),

Y G, AG, ^ ,^^

x=GrG;G=*^- ^^^"

.-.tan. a=taii. A6G„ .■.^=AGG,.

The inclination of the resultant E to Aa is therefore
equal to the angle AGG„ but the perpendicular to AG is
evidently inclined to Aic at this same angle. Therefore the
direction of the resultant R is perpendiciilar to the hue AG,
drawn from the axis to the centre of gi-avity. Moreover

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96 THE CENTRE OF 0801IXATIOH.

its magnitude and tine distanco of its point of application
from A liave been before determined by equations (83)
and (84).

The Centre of Pjcecussjon.

109. It is evident, that if at a point of the body througli
wliich the remdtant of the effective forces upon it pa^es,
there be oppwed an obstacle to its motion, then there will
be produced upon that obstacle the same effect as though
the whole of the effective forces were collected in that
point, and made to act there upon the obstacle, so that the
whole of these forces will take effect upon the obstacle, and
there will be no effect of these forces produced else-
where, and therefore no repercussion upon the axis.
It is for this reason that the point O in the resultant,
where it cuts the line AG drawn from the axis to the
centre of gravity, is called the centre of pekcussiok.
Its distance L from A b determined by the equation

■ (86),

which is obtained from equation (84) by writing MB? for I
(Art. 80.), K being the radius of gyration. If at the centre
of percussion the body receive an impulse when at rest,
then since the resultant of the effective forces thereby pro-
duced will have its direction through the point where the
impulse ts communicated, It follows that the whole impulse
■will take effect in the production of tliose effective forces,
and no portion be expended on the axis.

Tub Centre of Oscillatiok.

110. It has been shown (Art. 98.) that in the simple pen-
duhim, supposed to be a single exceedingly small element
of matter suspended by a thread without weight, the time
of each oscUlation is dependent upon the length of this
thread, or the distance of tlie suspended element from the
axis about which it oscillates. If therefore we imagine a
number of such elements to be tlius suspended at d^erent
distances from the same axis, and if we suppose them, after
having been at first united into a continuous body, placed
in an i^nclined position, all to be released at once from this

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THIS CENTRE OF OBCILLAllOH. 97

union with one another, and allowed to oacillsite ffeely, it is
manifest that; their oscillations will all be pei-foi-med in
different times. Kow let all these elements again be con-
eeiTed united in one osciliating mass. All being then com-
pelled to perform these oscillations in the same time, whilst
all tend to perfonn them in different times, the motions of
some ai-e manifestly retarded by their connexion with the
rest, and those of others aooelerated, the former being those
which lie near to the axis, and the others those more remote ;
so that ietwem, the two there must be some point in the
body where the elements cease to be retarded and begin to
be aocelerated, and where therefore they are neither accele-
rated nor retarded by their connexion with the rest ; an ele-
ment there performiag its oecillations precisely in the same
time as it would do, it it were not connected with the rest,
but suspended freely from the axis by a thi'ead without
weight. This point in the body, at the distance of which
from the axis a single particle, suspended freely, would per-
form its oscillations precisely in the same time that the body
does, is called the cbntke of oscillation.

The centre of osoillation poineides with the centre of

111. Por (by equation 79) the increment of angular velo-
city per second/ of a body revolving about an hori-
zontal axis, the forces impressed upon it being the
weights of its parts only, is represented by the for-
mula ^-^sin. ^, where ^ is the inclination to the ver-
tical of the line AG, drawn from the axis to its
centre of gi'avity. But (by eq^uation 84), L=^rfj^, where L

is the distance AO of the centre of percussion from the
axis,

■■■/!'=? sin. s

Now it has been shown (Art. 98.), that the impressed

moving force on a particle whose weight is w, suspended

from a thread without weight, inclined to the vertical at an

angle S, is represented by w sin. A ; moreover i(f represent

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38 THE CENTSE OF OSCULATION.

the incroment of velocity per eecond on thie paitiele, then

~-f is the effective force upon it. Tlierefore by D'Alein-

Bert'e principle,

»m.<=^, .•./=j,m.(, .■./=/L.

Now yL is the increment of velocity at tlie centre of
percussion, and /' is that upon a single particle suspended
freely at any distance from the axis. If such a particle
were therefore suspended at a distance from the axis equal
to that of the centre of percussion, since it would receive,
1^ the sa/me disiemce from the axis, the same increments of
velocity per second that the centre of percussion does, it
would manif^tly move exactly as that point does, and per-
form its oscillations in the same time tnat the body does.
Therefore, &c.

112. The centres of sii^pension and oscillation are red-

Let 0 represent the centre of oscillation of a body
when suspended from the axis A ; also let G be its
centre of gravity. Let AO=L, AG=&, OG— G-,;
also let the radius of gyration about A be repre-
sented by K', and that about G by ^. Therefore
(equation 59), K'— 0"+^ ;

, (equation 85), L= — ^=G + -^ (87),

.■.G-|-G,=G-f^,
.■.G.=| (88).

Now let the body be suspended from O instead of A ;
when thus suspended it will have, as before, a centre of
oscillation. Let tlie distance of this centre of oscillation
from O be L„

■'■ ^y equation (87), L,^G,+7t-)

./Google

•'• by equation (88), l.,=-^ + Q=li.

Since then the centre of oscillation in this second ease is at
the distance L from 0, it is in A; what was before the
centre of suspension hae now therefore become the centre
of oscillation. Thne when the centre of oscillation is con-
verted into the centre of suspension, the centre of snspen-
aion is thereby converted into the centre of oscillation.
This is what is meant, when it is said that the centres of
oscillation and suspension are reciprocal.

113. To determine thepath cf a lody projected obUquekf
in vacuo.

: tlie whole time, T seconds, of the flight of the

body to any given point P

' of its path, to be divided

' into equal exceedingly small

intervals, represented by
iiT, and conceive the whole
effect of gravity upon the
projectile during each one
of these intervals to be col-
lected into a single impulse at the tennination of that inter-
val, so that there may be communicated to it at once, by
that single impulse, aU the additional velocity which is in
realitnf communicated to it by gravity at the different periods
of the small time aT.

Let AB be the space which the projectile would describe,
with its velocity of projection alone, m the tirst interval of
time; then will it be projected from B at the commence-
ment of the second interval of time in the direction AET
with a velocity which would alone carry it through the dis-
tance BK= AB in that interval of time ; whilst at the same
time it receives from the impulse of gravity a velocity such
as would alone carry it vertically through a space in uiat in-
terval of time which may be represented by BF, By reason
of these two impulses communicated togeiher, the body will
therefore describe in the second interval of time the di:^o-
nal BO of the parallelogram of which BK and BF are adja-

,y Google

100 PROJECTILES.

cent aides. At the commencement of the thii'd interval it
will therefore have arrived at C, and wiU be projected from
thence in the direction BOX, with a velocity wtdcb would
alone carry it through OX^BO in the third interval ; whilst
at the same time it receives an impulse from gravity com-
municating to it a velocity which would alone carry it
through a distance represented by CG=BF in that interval
of time. These two impulses together communicate there-
fore to it a velocity which carries it through CD in the thii-d
intei-val, and thus it is made to describe all the sides of the
polygon ABOD ... P in succession. Draw the vertical PT,
and produce AE, BG, CD, &c. to meet it in T, N, 0 . . .,
and produce G-0, HD, &c. to meet BT in K, L, &c.

Now, since BO is equal to OX, and CK is parallel to XL,
therefore KL is equal to BK or to AB.

Again, since CD is equal to DZ, and DL is parallel to ZM,
therefore LM is equal to KL or to AB ; and so of the rest.

If therefore there he n intervals of time equal to ^H, so
that there are n sides AB, BO, OB, &c. of the polygon, and
n divisions AB, BK, &c. of the line AT, then AT, =wAB and

:.'m={ti-l)'KG={n-X)W.
Similarly 0N=(n-3)0X, therefore N0^('/i-2)DX=
()j, — 2)BF ; and so of the remaining parts of TP.

I^ow these parte of TF are (w— 1) in number, therefore
TP=(n-l)BF-|-(«-2)BF + (»-3)BF-|- . . . {{n-l) termBJ;
orT'P={{n-l) + {n-^)+ . . . ]m.

Therefore, summing the series to {n—1) terms.
TP=|2(»-l)-(«-2)!("-=i) . BF,

Kow g represents tlie additional velocity which gi'avity
would communicate to the projectile in each second, if it
acted upon it alone. g&T is therefore the veloci^ which it
would communicate to it in each interval of aT seconds,
graT is therefore the velocity communicated to the body by
each of the impulses which it has been supposed to receive
from gravity.

./Google

FliOJ'EGnUES. XUJ

Now EF i3 the epaco through which it would be can-ied
n the time aT hy this velocity,

AIbo aT=-,

=ii^(n-l)^=^l-^)T=.

Kow this is true, however small may be the intervale of
time aT, aad therefore if they be infiiiitely small, that is, if
the impulses of gravity be suppoeed to follow one another at
infinitely small intervals, or if gi'avifcy be supposed to act, as
it really does, cordvnuoualy.

But if the intervals of time aT be infinitely small, then
the number n of these intervals which make up the whole
finite time T, must be infinitely great. Also when n is infi-
nitely great, -^0.

In the actual case, therefore, of a projectile eontinually
deflected by gravity, the vertical distance TP between the
tangent to its path at the point of projection, and its position
P alter the flight has continued T seconds, is represented by

the formula

TP=i^ (89).

Moreover AT=nAB, and AB is the space which the body
would describe uniformly with the velocity of projection in
the time aT, so that nAB is tlie space which it would de-
scribe in the time n . AT or T with that velocity. If there-
fore V equal the velocity of projection, then

AT=V . T . . . . (90) ;
BO that the position of the body after the time T is the same
as though it had moved throngh that time with the velocity
of ite projection alone, describing AT, and had then fallen
through the same time by the force of gravity a^'One, describ-
ing TP (see Art. 101.).

A

114. Let AM=x, MP=?/, angle of
projection TAM=a, velocity of projee-

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PEOJECTILES.

a: tan. ^-y=m:—MP=TF=^gT (91).

Sabstitating the value of T from the pieceding equation,

;.y=xt&-n. "— ^yi, ■ ■ 1^-

Let 11 he the height through which a hody nmat fall freely
hy gravity to accLuire the vdocity V, or the height due tc

liiat velocity ; then V'^S^H (Art. 47.), therefore 4H=— ;
therefore, hy substitution,

y=.x tan. a— — —-ts' (93),

115. To find the time of the flight of a projectile.

It has been shown (equation 91), that if T represent the
time in seconds of the flight to a point whose co-ordinatea
are x and y, then

■|^"=ictan, «— y, /.T'^- jictan. «—y\,
.■.T=/?^^te^^-r^ (93).

N'o'w, -=5^-=— nearly, :.T='^^JxtaTi. a—y nearly.

li the projectile descend again to the horizontal plane fron:
which it was projected, andT he the whole time of its flight;
and X its whole range upon the plane, then, since at tlie ex
piration of the timfe T, y=0 and ic=X,

.•.T=^-^Xlan.a=i^Xtan. « nearly.

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PBOJEOXILKS.

When the projectile attains its greatest horizontal range,
its height y above the horizontal plane
becomes 0, whilst the abscissa x of the
point P, which it has then reached in
its path, becomes X. Substituting
these values 0 and X, for y and x in
equation (92), we have 0=X tan. a—
X' eec." «

=411 sin. a COS. a.

. X=4H tan. '

.•.X=3Hsin. 2«.

■ (9i)-

K the body be projected at different angular elevations,
but with the same velocityj the horizontal range will be the

greatest when ain. 3tt is the greatest, or when 2«=g, or '^=t-

117. To find the greatest height -which a projectile will
attcdn in its flight if projected with a given velocity ^ a/nd
at a given inclination to the horizon.

T Multiplying both sides of equation

,^'1 (92) by4H COS.' a-, we have 4H cos.' «

, y:^iK COS." a. tan. a . x—3^=2H (2
cos, a sin. a) x—af^^^'H. sin. Set . x—s?.
Subtracting both sides of this equa-
. tion from H" sin." 2a, we have

."2^— 2H8in. i

. a; +55".

H'BUi"2a-4Hc05.'a..y
But sin." 2a=4 sin." a cos.V,

.-. 4H cos.' a jH sin." « -~y\ -- \B. sin. 2a— a;} '. . . . (95).

Now the second member of this equation is always a
positive quantity, being a square. Va% first member is
therefore always positive ; therefore H sin.' ^—y is always
positive. "Whence it follows that y can never exceed H
sin." a, BO that it attains its greatest possible value when it
equals H sin.' "-, a value which it manifestly attains when

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104 PEOJEOTILES.

the first member of the above equation vanielies, or when
ai=H sin. 2k, that is, when a> hecomee equal to lialf the
greatest horizontal range, as is apparent from the last pro-
position; so tliat the greatest height BD of the projectile
IS represented by H em,' a, a height which it attains when
AD equals half the horizontal range.

118. The path of aprojectile vnvaouois apardbola.

/l Let B be the highest point in the

y"^ j flight of the projectile, and BD its

freatcst height. Draw PM, perpen-
icalar to BD. Let BM,=®„ M,P

:. a^,=BD— M,D=BD— PM:=H sin.'a— y,
y,=DM=AM— AD=iB— H sin. 2a.
Substituting these values in equation (95),
y,'=4H COS." a . i», (96),

■which is the equation to a porabola whose vertex is in
B, whose axis coincides with BD, and whose parameter ie
4H ccffi." a.

The path of a projectile in va/yao is therefore a parabola,
whose vertex is at the liighest point attained by the pi'o-
jectile, and whose axis is vertical.

119, To find the, rwnge of a projectile wpon an imUned
plane.

Let B represent the range AP of a projectile upon an
, inclined plane AB, whose inclination is
I, Then H and a being taken to repre-
"■■•^ sent the same quantities as before, and
"■ """ X, y being the co-ordinates of P to the
horizontal azis AC, we have

iK=AM=AP COS. PAM=R cos. i,
2/=PM=AP sin. PAM=R sin. i.

Substituting tliese values of x and y in the general equa-
tion (92) of the projectile we liave

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rEOJEOTILES.

W COS.' I sec'

R sin. 1=11 cos, 1 tan. a lTf~" "

Dividing by E, multiplying by cos. «, aiid transposing
E COS.' I aeo. a

ill

—sin. I COS. a=sin. (a-

,, E^ffl-g^'^'}""-" (971.

cos. 1

]^ow sin. (2ii — i)— ein. i=siii. |a + (" — ')] — sin. |a — (a—
i)\ =2 sin. (a— i) cos. «.

Substituting this value of 3 sin. (« — i) cos. a in tlie pre-
ceding ecLuation, we bave

E=2HiEHl(?^=?iJl (98).

( COS.' 1 j ^ '

Now it is evident tliat if « be made to vary, i remaining
the same, E will attain its gi'eatest value when sin. (3a — i)
is greatest, that is when it equals unity, or when 2a — 1=

n, or when o.z=:-t+-. This, then, is the angle of elevation

corresponding to the great^t range, with a given velocity
upon an incHned plane whose inclination is j.

If in the preceding expression for the range we substitute

] n— (f*— ') { for a, the value of the expression will be found

to remain the same as it was before ; for sin. (2a— i) will, by
this substitution, become sin. |ff— 2(a— i)— if =sin. 5"— (2a
— i)}=8ui. (3a— i), Tlie value of K remains therefore the

same, whether the angle of elevation be a or s— C'^— ')■

And the projectile will range the same distance on the
plane, whether it be projected at one of these angles of
elevation or the other.

Let BAG be the inelination of the plane ou which the
projectile ranges, and AT the direc-
tion of proieetion. Take DAS equal
to BAT. Hen BAT=TAO-BAO
=a-i. And SAC^DAC-DAS=

.. 2-BAT=^-(«-0- The range AP
B therefore the same, whether TAG or SAG be the angle of

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106 CENTEIEUGAI. TOECE.

elevation, and therefore whetlier AT or AS be the direction
of projection.

Draw AE bisecting the angle EAD, then the angle EAO

=BAO+BAE=EAO+iBAD==.+i(^-.)=^+^.

The angle EAO is therefore that corresponding to the
grectiest range, and AE is the direction in which a body
should be projected to range the greatest distance on the
inclined plane AB.

It is evident that the directions of projection AS and AT,
which correspond to equal ranges, are equally inclined to
the direction AE con'esponding to the greatest range.

120. The velocity of aprcjectUe at diffarent points of its
faih. It has been shown (Art, 56.), that if a body move in
any curve acted upon by gravity, the work accumulated or
lost is the same as would be accumulated or lost, provided
the body, instead of moving in a curve, had moved in the
direction of gravity through a space equal to the vertical
projection of its curvilinear path.

Tims a projectile moving from A to P will accumulate or
lose a quantity of work, which is equal to that which it would
accumulate or lose, had it moved vei-tically from M to P, or
from P to K, PM being the projection of its path on the
direction of gravity. S'ow the work thus accumulated or
lost eqtials one hah' the difference between the 'vwes vvixs at
the commencement and termination of tlie motion.

Let V equal the velocity at A, and v equal the velocity at

"W"
■^- — v'. Moreover, the work
9

W . PM, therefore V"-t)'=3^MP. Let PM=7/,

:.v'^Y'-2gy (99),

which determines tlie velocity at,any point of the curve.

CiCNTEIFU&AL FOECE.

121. Let a body of small dimensions move in any curvi-

,y Google

CENTEIFUGAl FOKOE. 10)

linear patii AB, impoUed continually towarda
a given point S (called a centre of force) ty a
given force, whose amonnt, when the body
has reached the point P in its path, is repre-
*■ sented by F.* Let PQ be an exceedingly
small portion of the path of the body, and
conceive the force F to remain constant and
parallel to itself, whilst this portion of its path is being de-
scribed. Then, if PE be a tangent at P, and QK be drawn
parallel to SP, PE ia the space which the body would have
traversed in the time of describing PQ, if it had moved
with its velocity of projection from P alone, and had not been
attracted towards S, and E.Q or PT (QT being drawn paral-
lel to EP) is the space through which it would have fallen
by its atti'action towards S amte, or if it had not been pro-
jected at all from P.f Let v represent the velocity which
it would have acquired on this last supposition, when it
reached the point T. Therefore (Art. 66.), if w represent the
weight of the body,

FxPT=i-%".

Now the velocity u, which the body would have acqnired in
falHng through the distance PT by the action of the constant
Jorce F, is equal to chuhle that which would cause it to de
scribe the same distance wniformly in the same time.J

Representing therefore by Y the actual velocity of the
body in its path at P, we have

V PR' " 'PE*

Substituting this value of v in the preceding equation,

* The force here epoken of, and represented by F, is the moving force, or
preasuie on the body (see Art. 92.), and is therefore equal lo tbat pressure
which would juat sustain ila attraotton towards S.

■j- See Art. IIS. (equations S9 and 90) ; what is proved there of a body acted
upon by the force of gravity which is constant, and whose direction ia con-
ataotly parallel to Itself, is evidently true of any other constant force similarly
retaining a direction parallel to itself. To apply the same demonstration ti)
any such cose, we have only indeed to assume g to represent another number
than 32i.

i If / represent the additional velocity per second which F would com-
municate to the body, and ( the time of describmg M, then (Art. 44.)

«=/(; but (Art. 46.)PT=^'=(-^ji=|i; so that ^ is the Telocity witli

which PT would be described 'omforaily in the time t.

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108 CENTltDX'&AL IfOBOE.

g \Piy g (PE)'

INow let a circle PQY te described having a common tan
eent witt tlie curve AjB in the point P, and passing tkroagL
the point Q, Produce PS to intei-sect the circumference of
this circle in V, and join QV; then are the ti-iangleg PQY
and QPE similar, for the angle EQP is equal to the angle
QPY (QE and VP heing parallel), and the angle QPR is
equal to the angle QYP in the alternate segment of the cii--

cle. Therefore ^=pY ; therefore QE=£^'. Suhsti-
tTiting this value of QE in the last equation, we have

Now this is true, however much PQ may he diminished.
Let it be mfinitdy diminished, the supposed constant amount
and parallel direction of F will then coincide with the actual
case of a variable amount and inclination of that force, the

ratio .p^ will become a ratio of equality, and the circle

PQV will become the circle of curvature at P, and PV that
chord of the circle of curvature, which being drawn from P
passes through S. Let this chord of the circle of curvature
be represented by 0,

..,F=2-^ (100).

The force or pressure F thus determined is manifestly
exactly equal to that force by which the body tends in its
motion continually to fly from the centre S, and may there-
fore be called its centrifugal force. This term is, however,
generally limited in its application to the case of a body re-
volving m a cvrde, and to the force with which it tends to
recede from the centre of that circle ; or if applied to the
case of motion in any other curve, then it means the force
with which the body tends to recede from the centi-e of the
circle of curvatnre to its path at the point through which it
is, at any time, moving. When the body revolves in a cir-
cular path, the circle of curvature to the path at any one
point evidently coincides with it tliroughout, and the chord
of curvature becomes one of its diameters. Let the radius
of the circle which tlie body thus describes be repi
byE, thenC^SE;

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OEtTTEIFUGAL FOKCE. 109

■•■ F='^^ (101).

Since in whatever ciirve a body is moving, it maj be ton
ceived. at any point of its puih to be revolving in the circle
of cnrvatare to the curve at that point, the force F, witli
which it then tends to recede from the centre of the circle
of cui'vatm-e ia represented by tlie above formula, JR being
taken to represent the radius of owrvaiwre at the point of its
path through which it is moving.

If « he the angular velocity St the hody'a revolntion about
the centre of its circle of cui'vature, then V=cill ;

.■.F='^a'E (102).

I?

122. From equation (100) we obtain

^■=*(?)°=K?)tt«)-

second /, which woiild be communicated to a body falHng
towards S, if the body fell freely and the force F remained
constant. Moreover, by Art. 47. it appears, that V is the
whole velocity which the body would on this supposition
acquire, whilst it fell through a distance equal to it), or to
one quarter of the chord of curvature. Thus, then, the velo-
city of a body revolving in any curve and attracted towards
.a centre of force is, at any point of that curve, equal to that
which it would acquire in falling freely from that point to-
wards the centre of force through one quarter of that chord
of curvature which passes tln-ough the centre of force, if the
force wh/ioh anted wpon it at that point in the cmve re-
-'--'■' ■'—■ ' ^- - -'-■- -' -' Itisir '■^'- '■^-'

g its desoent. It is in this sense that
the velocity of a body moving in any curve about a centre
of force is said to be that ntra to one quaktbk the ohoed

OF CDEVATBE.

123. The centriftgal force of a mass of finite diin,ensi(ms.

Let BC represent a tliiu lamina or sHce

of such a mass contained between two planes

exceedingly near to one another, and both

pei-pendicular to a given axis A, about

„ which the mass is made to revolve.

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liU CENTKIFDGAL FOIiCE.

Through A di-OM any two recliaogulai- axes Ax and Ay,
let m, be any element of the lamina whose weight ia w^, and
let AM, and AN,, co-ordinates of wi„ he represented by le,
and y,. Then by equation (102), if a represent the angnlar
velocity of the revolution of the body, the centrifugal force
on the element m, is represented by — w,Am,. Let now thia

force, whose direction is Am, he resolved into two others,
whose directions are Aic and Ay. The former will be repre-
sentedhy— «i,Am,, COS. icAm,, or by-_w,ic„ and the latter

by— «',Ato, cos. yAm.,, or by— w,yj; and the centrifugal

forces and all the other elements of the lamina being simi-
larly resolved, we shall have obtained two sets of forces,
those of the one set being parallel to Aic, and represented

bv — w,*,, — «'JK,, — w!,a;,, &c. and those of the other set

ff ff ff , . ,

parallel to Aw represented hy — wa/^, —w,y^, — v\y^, &c.
a Q 9
Now if S and T represent the resolved parts parallel to
the directions of Aic and Ay, of the resultant of these two
Bets of forces, then (Art, 11.)

X=— wt,iB.+ — waM — 'iCA+ , , . =•— 2«i!e=— "WG,;
g 9 g 9 9

g '^' g '^" g ■ g " g

if G, and G, represent the co-ordinates AG, and AG, of the
centre of gravity G of the lamina, and W its weight
(Art. 18.).

Now the whole centiifugal force F on the lamina is the
resultant of these two sets of forces, and is represented by

= r ^W'G,' -I- ^W=G,^ = "■ W Vg,' + l.;, or

F^^^W.G (103),

9
where G is taken to represent the distance AG of the centre
of gravity of the lamina from the axis of revohition.
Horeover, the direction of tins resultant centrifugal force

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is tliroiigli A, since tlie direction of all its eomiionents are
through that point

124. From tlie above formula, it is apparent that if a body

revolving round a fixed axis be conceived to

"^ /^=^ be divided into laminee by planes pei'pendicu-

^■^V:^:^ lar to the axis, then the centrifugal force of

^^0.:^ each such lamina ia the same as it would

^^^^^ have been if the whole of its weight had

^^^ been collected in its centre of gravity ; so

o that if tlie centres of gravity of all the laminge

be in the same plane passing through the

axis, then, since the centrifugal force on each lamina has its

direction from the axis through the centi'e of gravity of that

lamina, it follows that all me centrifugal torces of these

laniinfe are in the same plane, and that they are pakjlllel

forces, so that their resultant is equal to their sum., those

being taken with a negative sign wliich correspond to

laminte whose centi-es of gravity are on the opposite aide of

the axis from the rest, and whose centrifugal forces are

therefore in the opposite directions to those of the rest.

Thus if F' represent the whole centrifugal force' of such a

mass, then F'= — SWG. Now let W represent the weight

of the whale mass, and G-' the distance of its centre of gra-
vity from the axis, therefore 2WG='W'Gr' ;

In the case, then, of a revolving body capable of being
divided into lamina perpendicular to the axis of revolution,
the centres of gravity of all of which laminse are in tlie
same plane passmg through the axis, the centrifugal force is
the same as it would have been if the whole weight of Hie
body had been collected in its centre of gravity, the same
property obtaining in this ease in respect to the wliole body-
as obtains in respect to each of its individual laminte.
Since, moreover, the centrifugal forces upon the laminge are
parallel forces when their centres of gravity are all in the
same plane passing through the axis of gravity, and since
their directions are all in that plane, it follows (Art. 16,),
that if we take any point 0 in the axis, and measure the
moments of these parallel forces from that point, and call
V the perpendicular distance OA of any lamina BC from

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113 THE PKINCIPLE OF TIETtlAL VELOCITIEe.

that point, and H the distance of their resultant from tliat
poiEt, then

g g '

(105).

The equations (104) and (105) determine the amount and
the point of application of the resultant of the centrifugal
forces upon the mass, upon the supposition that it can oe
divided mto lamina perpendicular to the axis of revolvition,
all of which have their centres of gravity in the same plane
passing through the axis.

It is evident that this condition is satisfied, if tlie body be
symmetrical as to a certain axis, and that axis he in the
same plane "witli the axis of revolution, and therefore if it
intersect or if it be parallel to the axis of revolution.

If, in the case we Imve swpjposed, 2"W"G=0, that is, if the
centre of gi'avity he m, the axis of revolution, ihen the cen-
trifugal force vanishes. This is evidently the case ■where a
body revolves round its axis of symmetiy.

[ gravity o

the body is divided by planes perpendicular to
the axis of revolution be not in the same plane
(as in the figure), then the centrifugal forces of
the different laminfe wiU not lie in the same
plane, hut diverge from the axis iu different
directions round it. The amount and direction
of their resultant cannot in this case be deter-
mined by the equations which have been given
above.

The PiuNCiPLE of viktual Velocities.

126. If any presswre P, vihrne jyoint of ajjpli..
■mtms to move through the straight tine AB,
mto three others X, i, Z, in the dn/rections oT the three
rectamginlaT asms. Ox, Oy, Os; cmd if AC, AD, and AE,
he the projections of AB upon these axes, then the work of
P thrmigh, AE is egual to the swn of the works of X, Y, Z,
through AC, AD, and AE respectively, or X . AO+T .

AS + Z. AE=P. A¥.

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TIIE PEINOIPU: OF VIKTCAL YELOCITIEB.

Let the inclinations of the direction
of P to the axes On;, Oj', Os respec-
tively, be represented by a, /3, -y, and
the mclniatione of AB to the Bams

axes by «„ .S,, 7,,

.-. (Art. 12.) X=P COS. a, Y=P cos. ,3, Z=P cos. y; also AG

=AB CO8. «„ AD=AB COS. ,3„ AE=AB cos. Ti.
.-.X. AO=P. AB COS. « COS. «„ Y.AD=P. AB cos. /3 cos. 3„

Z . AE=P . AB cos. y cos. y,,

:.X . AC+y . AD + Z . AE=P . AB jcos. » cos. s + eos. ^

COS. /3| + cos. y COS. y,}.

But by a ivell-lniown theorem of ti'iaonometry, cos. a cos.

a, + cos. /3 COS. /3, + cos. y cos. y,=c08. PAB,

.-.X . AC+T .1^ + 2. AE=P . AB cos. PAE;
bntABcos. PAB=:AM;

.-.X.AC+Y. AD + Z. AE=P. Ail.

But (Art. 52.) the work of P through AM is equal to its
work through AB. Therefore, &c.*

127. If a/tvy wwrriber of foreea Je m egyMibriMm Q>emg m.
am/ iva/y me(ihaima(My oonneeied with one anoih&r), and */",
swjeet to thai cormeeidon, thmr d/ijf&rerd pomts of wp^i-
cabwn he mads to move, meh through any exoeedrngly smaU
distance, then the aggregate of the work of those forces,
■whose foimis of a^ticabum a/re made to move towards the

* This proportion may readily be deduced from Art. 63., for pressurae equal
and oppOfflte to X, Y, Z, would juat iie in equilibrium ivith P, and theae tend-
ing to move the point A in one direction along the line AB, P tenda to move
il, m the oppoate direotion, therefore in the motion of the point A through AB,
the aura of the works of X, Y, Z, rauat equal the work of P. But the work of
S, as its point of application moTea through AB, ia equal (Art. 62.) to the
work of X through the projection of AB upon As, that ia, through AC ; aimi-
larlj the work of T, aa its point of applieation moves through AB, is equal i«
its work tlirough the projection of AB upon Ay, or through AD ; and ao of Z.
The sum of the worka of X, T, and Z, aa their point of application is made to
move through AB, is therefore equal to what would have been the aum of their
works had their points of application been made to more separately through
AC, AD, AE 1 this last sum is therefore equal to the work of P through AB,
which ia equal to the work of P through AM, AM being the projectlou of AB

the direction of P.

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114 PEINCIFLB OF VIKTUAL VELOCITIES.

directions in, which, the several forces appUed to tlv&m act_
shall equal the aggregate of the worh of those forces, the
motions of whose vomts of (^Uoaiion iwe opposed to th6
dk-6ctions of the forces apipUm to them.

For let all the forces composing snch a eystein be re-
solved into three sets of forces parallel to three rectangular
axes, and let these three sets of parallel forces be repre-
sented by A, B, and C respectively. Then must the result-
ant of the parallel forces of each set equal nothing. Por if
any of these resultants had a finite value, then (by Art. 12.)
the whole three sets of forces would have a resultant, which
they cannot, since they are in equilibiinm.

Kow let the motion of the points of application of the
forces be conceived so smtM that the amounts and dwections
of the forces may be made to vary, during the motion, only
by an exceedingly small quantity, and so that the resolved
forces upon any point of application may remain sensibly
unchjuiged. Also let m,, «„ ■m^, represent the works of these
resolved forces respectively on any point, and 2w, the sum
of all the works of the resolved forces of the set A, ^u^ the
sura of all the works of the forces of the set B, and 2k, of the
set C, l^ow since the parallel forces of the set A have no
resultant, therefore (Art. 59.) the sum of the worts of tbose
forces of this set, whc«e points of application are moved
towards the directions of then- forces, is equal ia the sum of
the works of those whose points of application are moved
from the directions of their forces, so that 2i(,=0, if the
values of 1*,, which compose this sum, be taken with the
positive or negative sign, according to the last mentioned
condition.

Similai-ly, 2m,=0 and 2'w,=0, ,'. s('W,+-m,+'W^^O,
I^'ow let tJ represent the actual work of that force P„ the
works of whose components parallel to the three axes ai'e
represented by Wj, w^, m, ; then by the last proposition,

.-. 2U=0 (106);

in which expi'easion U is to be taken positively or n
according to the same condition as w„ u„ u, ; that ii
ing as the work at each point is done in the direction of the
corresponding force, or in a direction opposite to it. Hence
therefore it follows, from tlie above eqiiations, that the sum

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of tlie works in one of these directions equals tlieir sum in
the opposite direction. Therefore, &c.

The projection of the line descrihed hy the point of appli-
cation of any force upon the direction of that force is called
its viETiTAL VELOcrrY, 80 that the product of the force by ita
virtual Telocity is in fact the work of that force; hence
therefore, representing any force of the system by P, and
its virtual velocity by p, we have Pp=U, and therefore,
2Pp=0, which is the principle of virtual velocities.*

128. ^ there he a system of foToes such that th&ji' povnts of
mjAieation heing moved through ceHcdn consecutive ^si-
twnSf those forces are m all such poaitdons m eqmUhrvmn,,
then m retpeat to a/mjfmte motion of the points of am>U-
oaHon through that series of positions, the ag^egc^e of the
worh of those forces, which act m the directtons in which
their several points of appUaation a/re made to move, i '

'e of the worh in the q

This principle has been proved in the preceding proposi-
tion, only when the motions communicated to the several
points of application are exceedingly small, so that the work
done by each force is done only through an exceedingly
small space. It extends, however, to the case in which each
point of application is made to move, and the work of each
force to be done, throngh any distance, however great, pro-
vided only that in all the different positions which the points
of application of the forces of the system are thus made to
take up, these forces be in eqnilibrinm witli one another ; for
it is evident that if there be a series of such positions
immediately adjacent to one another, then the principle
obtains in respect to each small motion from one ot this set
of positions into the adjacent one, and therefore in respect
to the sum of all such small motions as may take place in the
system in its passage from am/ one position into any other,
that is, in respect to the whole motion of the system throngh
the intervening series of positions. Therefore, &c.

The Pbihciple of Vis Viva,

129. If the forces of amy system he not m e^idUbrmm with
one another, then the difference "between the aggregate wcrh

' This proof of the principle of yirtiial velocities is given liere for ths first

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116 THE PEIKCIPLE OF TTS YTfA.

of those whose tendency is in the direction of the motions
of thei/r several p&kits of (wplication, and those wJiose ten-
denoy is i/n the opposite dwedion, is equal to one half the
aggregate vis vima of the system.

In eacTi of the consecutive positions wbicii the bodies com-
uosing the system are made successively to take up, let there
be applied to each body a force equal to the effeeti/oe force
(Art. 103.) upon that body, but in an opposite direction;
every position will then become one of equilibrium,

Tfow, as the bodies which compose the system and the
various points of application of the improved forces, move
through any finite distances from one position into another,
let Sti, represent the aggi'egate worE of those impressed
forces whose directions are towards the directions of the
motions of their several points of application, and let Sm,
represent the work of those impressed tovcea which act in the
opposite directions ; also let S«, represent the aggregate
work of forces applied to the system equal and opposite to
the effective forces upon it; the directions of these forces
opposite to the effective forces are manifestly opposite also
to the directions of the motions of their several points of
application, so that on the whole S«, is the aggregate work
of thcee forces whose directions are towards the motions of
their several points of application, and Sm,+S«, the aggre-

fate work opposed to them. Since therefore, by D'Alem-
ert's principle, an equilibrium obtains in every consecntive
position of the system, it follows by the last proposition,
that

.-. 2m,— Su,= Si(, (107).

INow Mj is taken to represent the work of a force equal and
opposite to the effective force upon any body of the system ;
but the work of such a force through any space is equal to
the work which the effective force (being unoppcBed) accu-
midates in the body through that space (Art. 69^, or it is
equal to one half the difference of the vires vivte of the body
at the commencement and teimination of the time during
which that space is described (Art. 67.). Therefore SWj
equals one half the aggregate difference of the vi/res imm of
the system at the two periods ;

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VITA. 117

Thus then it follows, that the difference between the aggre
gate work lu, of tliose forces, the t«udeiicy of each of which
IS towards ihe direction of the motion of its point of applica-
tion, and that Sm, of tliose the direction of each of which is
opposed to iJie motion of its point of application (or, in other
words the difference between the aggregate work of the
aocderaimg forces of the system aad tuat of the retarding
forces), is equal to one half the vis viva accumulated or lost
m the system whilst tlie work is being done, which is the
PRraciPi.E OF Vis Viva.

130, One half the vis viva of tlie system i
accumulated work; the principle of vis viva amounts,
therefore, to no more than this, that the entire difference
between the work done by those forces which tend to accele-
rate the motions of the parts of the system to which they
are applied, and those which tend to retard them, is clcou-
mulated in the moving parts of the system, no work
whatever being lost, but alt that accumulated which is don;',
upon it by the forces which tend to accelerate its motion,
above that which is expended upon the retarding forces,

This principle has been proved generally of any mechani-
cal system ; it is therefore true of the most complicate,!
machine. The entire amount of work done by the movin '■
power, whatever it may be, upon that machine, is yieldel
partly at its working points in overcoming the resistancad
opposed there to its motion (that is, in doing its useful
■work), it is partly expended in ovei-coming the friction and
other prejudicial resistances opposed to the motion of the
machine between its moving and its working points, and all
the rest is aocivrrmlated in the moving parts of the machine,
ready to be yielded up under any deficiency of the moving
power, or to carry on the machine for a time, should tho
operation of that power be withdrawn.

131. Wh&n, the forces cf (my ayst&m {not in, equilibrium in.
ffoery portion wMoh the pa/rts of that system may he
Tfiade to take 'wp) pass through a position of equiZilyrium,
the vis viva of the system, hecomes a inaximum or a
mm/t/mwrn.

For, as in Art. 139., let the aggregate work done in the
directions of the motions of the several parts of tlie system

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118 PosrrioH of maximum oil mihimum: vis tiva.

be repreeented by ^u^, and the aggregate work done in
directions opposed to the motions ot the eevera! parts by
Sw,, then (Art. 129.), one half the aeqmi-ed vie viva of
eyBteiii=2i/j— 2mj. Kow as the system pa^es from any one
position to any other, each of the C[uantiti^ 2w^ and ^u, is
manifestly increased. If Su^ increases by a greater q^uan-
tity than 2^^, then the vis viva is increased in this change
of petition ; if, on the contrary, it is increased by a less
quantity than 2«^, then the vis viva is diminished. Thus if
ASMj and a2Mj represent the incremente of s-m^ and Sm^ in
this change of position, then {^u^+'i'Su^—{su,+ASu,), or
(2u,— 2t*j) + (A2'M|— A2«A representing one half the vis
vivi after this change ot position, it is manifest that the vis
viva is increased or diminished by the change according as
AS«, is greater or less than as-m^ ; and tliat ii' a2«, be equal
to a2Mj then no change takes place in the amount of the
vis viva of the system as it passes from the one position to
the other.

Now from the principle of virtual velocities {Art. 127.),
it appears, that precisely this case occurs as the system
pa^es through a position of equilibrium, the aggregate
work of those forces whose tendency is to accelerate tlie
motions of their points of application then precisely equal-
ling that of the forces whose tendency is opposed to these
motions. For an exceeding small change ot position there-
fore, passing through a position of equilibrium, A2t*=A2M^,
an equahty whicli does not, on the other hand, obtain,
unless the body do thus pa^ through a position of equili-
brium.

Since then the sum 2w, — Sw„ and therefore the aggregate
vis viva of the system, continually increases or diminishes
up to a position of equilibrium, and then ceases (for a cer-
tain finite space at least) to increase or diminish, it follows,
that it is m that position a maximum or a minimum.
Therefore, &e.

132. When the forces of (my systempass through aposition
of eqmMhrinem, the vis ■vwa becomes a mamrnvm or a
fmrn/mwrn.^ accordi/ng as thai position is one of stable or
■unstable egmUhri/um.

For it is clear that if the vis viva be a maximum in any
position of the eqnihbrimn of the system, so that after it
has passed out of that position into another at some finite

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STABLE AND UKSTABLE EQUILIBBHIM. 113

distance from it, the acquired via viva may have become
less than it was before, then the aggregate work of the
tbrces which tend to accelerate the motion between these
two positions mnBt have been less than that of the fore^
which tend to retard the motion (Art. 131.). Now suppose
the body to have been placed at rest in this position of
eqTiilil)rium, and a small impulse to have been communi-
cated to it, whence has resiilted an aggregate amount of
vie viva represented by SniV. In the transition from the
first to the second position, let this vis viva have become
Stm)' ; also let the aggi'egate work of the forces which have
tended to accelerate the motion, between the two positions,
be represented by 2U„ and that of the forces which have
tended to retard the motion by 2U, ; then, for the reasons
assigned above, it appears that 2U5 is greater than sU,.
Moreover, by the principle of vis viva,

^2^' _ ^2mV' =SU, -SF„
;, 2>m)'=s«iV'— 2(2U,— 2U,)-,

in which equation the quantity 3(2TJs — 2U,) is essentially
positive, in respect to the particular' range of positions
through which the disturbance is supposed to take place.*

For evei-y one of these positions there must then be a
certain valiie of 2m V, that is, a certain original impulse
and disturbance of the system from its position of equili-
brium, which will cause the second mejiiber of the above
equation, and therefore its first member ^r/w", to vanish.
Now every term of the sum 2«j^' is e^entially positive ;
this sum cannot therefore vanish unless each term of it
vanish, that is, unless the velocity of each body of the
system vanishes, or the whole be brought to rest. This
repose of the system can, however, only be instantaneous ;
for, by snpposition, the position into which it has been dis-
placed is not' one of equilibrium. Moreover, the displace-
ment of the system cannot be continued in the direction in
which it has hitherto taken place, for the negative term in
the second member of the above equation would yet farther
be increased so as to exceed the positive term, and the first

* The dieturbanca is of course to be limited to that particular range of
poaltiona to which the supposed portion of equilibrium stands in the relation
of a poation of maximum via vIti. If there ba othar portions of eqnili-
hrium of the system, there will be other ranges of adjacent positions, in
respect to each of which there obtains a similar relation of masimum or mini'

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i20 8TAELE AND USSTjiBLE EQUILIBRIUM.

memljer "s^mv^ ■would thus become negative, wliicli is
irapoBsible.

The motion of the system can then only be continued by
the directions of the motions of certain of the elements
whicli compose it being changed, bo that the corresponding
quantiti^ by which 2U, and 2Uj are respectively increased
may change their signs, and the whole quantity sTJ, — sU,
which before m<yreased continually may now continually
dmwmsh. This being the case, Sm/a" wiU increase again
until, when 2TT, — SU5=0, it becomes again equal to SmV^ ■
that is, until the system acquires again the vis viva with
which its disturbance commenced.

Thus, then, it hae been shown, that in respect to every
one of the supposed positions of the system* there is a cei--
tain impulse or amount of vis viva, which being communi-
cated to the system when in equilibrium, will just cause it
to oscillate as far as that position, remain for an instant at
rest in it, then return again towards its position of equili-
brium, and re-acquire the vis viva with which its displace-
ment commenced. Now this being true of every position
of the supposed range of positions, it follows that it is true
of every disturbance or impulee which will not carry the
system beyond this supposed range of positions ; so that,
having been displaced by any such distm'bance or impulse,
the system will constantly return again towards the position
of equilibrium from which it set out, and is stabile in
respect to that position.

On the other hand, if the supposed position of equOi-
biium be one in which the vis viva is a minimum, then the
aggregate work of the forces which tend to accelerate the
motion must, after the system has passed through that posi-
tion, exceed that of the forces which tend to retard the
motion ; so that, adopting the same notation as before, 2U,
must be gi'eater than SU,, and the second member of the
equation essentially positive. Whatever may have been the
original impulse, and the communicated vis viva 2mV,
Sjjiu' must therefore continually increase ; so that the whole
system can never come to a position of instantaneous repose ;t
but on the contraiy, the motions of its parts must continu-
ously increase, and it must deviate continually farther from
its position of equilibrium, in which position it can never

' That is, of that range of poaidona oyer whiuh the supposed poaition of
eqnillbrium holds the relation of a poaition of maximum t!s yiTa.

I Within that range of poaitiona over which the supposed position of
equilibrium holds the relation of minimum tIs viva.

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DYNAMICAL

rest. The position is tlius oiio of iiostalile oqiiilibiium.
Therefore. Ac.

Dynamical Stabilfiy.*

If a body be made, by the action of certain disturbing
forces, to pass from one position of equilibrium into another,
and if in each of the intermediate positiona these forces are
in excess of the forces oppc^ed to its motion, it ie obviotia
that, by reason of this excess, the motion will be continually
accelerated, and that the body will reach its second position
with a certain finite velocity, whose effect (measured under
the form of vis viva) will be to carry it beyond that position.
This however passed, the case will be reversed, the resist-
ances will be in excess of the moving forces, and the body's
veloci^ being continually diminished and eventually de-
stroyed, it wnl, after resting for an instant, a^ain return
towards the position of equiubrinm through which it had
passed. It will not however finally rest in this position until
it has completed other oscillations about it. Now the am-
plitude of the first oscillation of the body beyond the posi-
tion in which it is finally to rest, being its greatest ampli-
tude of oscillation, involves praetacally an important condi-
tion of its stabihty; for it may be an amplitude sufficient to
caiTy the body into its next adjacent position of ecLuilibrium,
which being, of necessity, a position of unstable ecLuibbrium,
the motion wiU he yet farther continued and the body
overturned. Different bodies requiring moreover different
amounts of work to be done upon them to produce in all the
same amplitude of oscillation, that is (relatively to that ■ am-
plitude) the most stable which requires the greatest amount
of work to be so done upon it. It is this condition of stabi-
lity, dependent upon dynamical considerations, to which, in
the following paper, ttie name of dynamical stability is
given.

I cannot find that the qilestion has before been considered
in this point of view, but only in that which determines
whether any given position be one of stable, unstable, or
mixed equilibrium ; or which determines what pressure is
necessary to retain the body at any given inclination from
such a position.

• Istraoted from a paper " On Dynamical Stability, and on the Oscillations
of Floating Bodiee," by the author of tbia work, published in the Transactiona
of the Royal Society, Pact. II. for 1850. The remainder of the paper will bt
found in the Appendix.

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192

1, To the discussion of the conditions of the dynamicaj
stability of a body the principle of vis vwa readily lends
iteelf. That piinciple,* when translated into a language
■which the lahours of M. Pohoelbt have made familiar to
the uses of practical science, may he stated as follows;—

""Wlien, being acted npon by given forces, a body or sys-
tem of bodies has been moved from a state of rest, the difter-
ence between the aggregate work of those forces whose
tendencies are in the directions in which their points of
application have been moved, and that of the forces whose
tendencies are in the opposite direction, is ecLual to one-half
the vis vi/oa of the system."

Thus, if 2«, he taken to represent the aggregate work of
the forces by which a body has been displaced from a posi-
tion in which it was at rest, and 2w, tne aggregate work
(duiing this displacement) of tbe other forces applied to it ;
and if the terms which compose 2m, and 2m, be understood
to be taken positively or negatively, according as the ten-
dencies of the corresponding forces are in the directions in
which their points ot application have been made to move
or in the opposite. directions; then representing the aggi-e-

gate vis viva of the body by - ^wv^.

2k,-|-SMj= „- sww', (1').

IlTow 2mj representing the aggregate work of those forces
which acted upon the body in the position from which it has
been moved, may be supposed to the known; ^u^ may there-
fore be determined in terms of the vis vima, or conversely.

3. In the extreme position into which the body is made to
oscillate and from which it begins to return, it, for an instant,
rests. In this position, therefore, its vis vi/oa disappeare, and
we have «

s-m,-1-2m,=0 (2').

This eCLuation, in which Xu, and ^u, are functions of the
impressed forces and of the inclination, detenuines the ex-
treme position into which the body is made to roll bv the
action of given distm-bing forces ; or, conversely, it deter-
mines the forces by which it may be made to roll into a
given extreme position.

• See Art. 129.

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DTNAMIOAL STABILITr. 133

3. The position in which it will finally rest is determined
by the maximum value of s^i + 2m, in equation (!') ; for, by
a well-known property, the vis tma of a system* attains a
maximiun value when it passes thi'oush a position of stable,
and a minimum, when it passes through a position of unstable
equilibrium. The extreme position into which the body
oscillates is therefore essentially diffei'ent from tliat in wliieh
it will finally rest.

4. Different bodies, requiring different amounts of work to
be done upon them to bring them to the same given inchaa-
tion, that is (relatively to mat inclination) the most stable
which requires the greatest amount of work to be so done
upon it, or in respect to which Su, is the gi-eateet. If, in-
stead of an being brought to the same given inehnation, each
is brought into a position of unstable equilibrium, the coiTe-
sponding valae of 2«, represents the amount of work which
must be done upon it to overthrow it, and may be considered
to measiire its absolute, as the fonner value measures its
relatwe dynamical stability-f The absolute dynamical sta-
bility of a body thi^ meaanred I propose to represent by the
symbol U, and its relative dynamical stability, aa to the

' inclination 6, by U(fl).

The measure of me absolute dynamical stability of a body

the maximum value of its relative stability, or U the max-

mm of U(d) ; for whilst the body is made to incline from

i position of stable equilibrium, it continually tends to

return to it until it passes through a position of iinstable

equihbrium, when it tends to recede fi'om it ; the aggregate

amount of work neeeasaiy to produce this inclination must

therefore continually increase until it passes through that

position and afterwards diminish,

5. The work opposed by the weight of a body to any
change in its position is measured by the product of the
vertical elevation of its centre of gravity by its weight.1
Representing ther^ore by W the weight of the body, and
by aH the vertical displacement of its centre of gravity
when it is made to incline through an angle fl, and observ-
ing that the displacement of this point is in a direction oppo-
site to that in which the force applied to it acts, we tiava
2k,= — "W.aH, and by equation (2'),

» Art. 132.

fit is obrioua that the absolnte dyoamioal stability of a body may be
greater than that of another, whilst its stability, relatirely to a giren inelina.
tion, is less ; less wofk beinj^ required to incline it than the other at thai
angle, but more, entirely to overthrow it.

I Art. 60.

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U(d)-W.AH=0 (8).

If therefore no otlier force than itB weight be opposed io a
body's being overthrown, its absolute dynamical stability,
when resting on a rigid surface, ia measiired hj theprodmci
of its weight ly the height through which its centre ofgra/oity
tmist te -raised to iring it from a stable into an wistable
position of eqiMlibrimn.

6. The JJynamicdl StahUity of Floatwig Bodies. — ^The
action of gusts of wind upon a ship, or of blows of the sea,
being measured in their effects upon it by their work, that
vessel is the most stable under the influence of these, or will
roll and pitch the least (other things being the same), ■which
requires the greatest amount of worh to be done upon it to
hnn^ it to a given inclination ; or, in respect to which the
relative dynamical stability TT (f) is the greatest for a given
value of ». In another sense, that ship may be said, to be the
moststablewhichwould require the greatest amount of work
to be done upon it to briog it into a position from which it
would not again right itself, or whose absolute dynamical
stability TJ is the greatest. Subject to the one condition,
the ship will roll the least, and subject to the other, it will
be the least likely to roll over.

Thus the theory of dynamical stability involves a question
of naval construction. It will be found diseu^ed in its ap-
plication to this question in the Appendix.

rEICTION.

1S3. It is a matter of constant experience, that a certain
resistance is opposed to the motion of one body on the sur-
face of another under any pressure, however smooth may he
the surfaces of contact, not only at the first commencement,
but at every subsequent period of the motion ; so that, not
only is the exertion of a certain force necessary to cause tlie
one body to pass at first from a state of rest to a state of mo-
tion upon the smface of the other, hut that a certain force is
further requisite to heep wp this state of motion, The resist-
ance tbiis opposed to the motion of one body on the surface
of another ■srlien the two are j^j'ess^f? together, is called fric-

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FEionoN. 125

tion ; that which opposes itself to the transition from a state
of continued rest to a state of motion is called tlie friction,
((f gmeamriGe ; that which continually aooompanies the state
of motion is called the frioUon of motion.

The principal expenments on Mction have been made by
Ooulomb*, Vince, Q-. Eennief, N. "WoodJ, and recently
(at the expense of the French Government) by Morin.g
They have reference, first, to the relation of tlie friction
of quiescence to the ij-iction of motion ; secondly, to the
variation of the friction of the same surfaces of contact undei
differmd pressures; thirdly, to the relation of the friction to
the esctent of tlie surface of contact ; fourthly, to the relation
of the amount of the friction of motion to the velocity of the
motion ; fifthly, to the infiaence of imguents on the laws of
friction, and on its amoimt under the same circumstances of
pressure and contact. The following are the principal facts
which have resulted from th^e experiments ; they consti-
tute the loAos of friction.

1st. That the friction of motion is subject to the same
laws with the friction of quiescence (about to be stated), but
agrees with them more accurately. That, under the same
circumstances of pre^vire and contact, it is nevertheless dif-
ferent in amount.

Sndly. That, when no unguent is interposed, the friction
of any two surfaces (whether of quiescence or of motion) is
directly proportional to the force with which they ai'e pressed
perpendicularly together (up to a certain ImhU of that pres-
Bui'e per square inch), so that, for any two ^ven surfaces
of contact, there is a constant ratio of the friction to the per-
pendicular pressm'e of the one sui'face upon the other.
Whilst this ratio is thus the same for the same surfaces of
contact, it is different for different suifaees of contact. The
particular value of it in respect to any two given surfaces
of contact is called the CO-EFFICIENT of friction in re-
spect to those surfaces. The co-efficients of fi-iction in respect
to those surfaces of contact, which for the most part form the
moving surfaces in machinery, are collected in a table, which
will be foimd at the teimination of Art. 140.

3rdly. That, when no imguent is intei-posed, the amount
of the friction is, in every ease, wholly independent of the
extent of the surfaces of contact, so that the force with which
two surfaces are pressed together being the same, and

" M&n, flea Sar. Btrang. 1181, + Phil, Trims, 1829,

t A Practical TreatlsB on Sail-roads, 3il ed. chap. 16.
I M^m. lie rinstitut, 1SS3, 1834, 1338.

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126 FEionoN.

not exceeding a certain limit (per square iiicli), tlieir friction
ie the same whatever may be tlie extent of their surfaces of
contact.

4thly. That the friction of motion is wholly independent
of the velocity of the motion.*

5t)ily. That where unguents are intei-poeed, the co-efficient
of friction depentb iipon the nature of the unguent, and upon
the greater or lees abundance of the supply. In respect to
the enppiy of the lingent, there are two extreme cases, that
in which the surfaccB of contact are but slightly rubbed with
the unctuous matterf, and that in which, by reason of the
abudant supply of the unguent, its viscous cousistency, and
the extent of the surfaces of contact in relation to the insist-
ent pressure, a continuous stratum of unguent remains con-
tinually interp(»ed between the moving surfaces, and the
friction ie thereby diminished, as far as it is capable of being
diminished, by the interposition of the pai-ticular unguent
used. In iJiis state the amount of Motion is found (as might
be expected) to be dependent rather upon the nature of the
unguent than upon that of the surfaces of contact ; accord-
ingly M. Morin, from the comparison of a great number of
reeiitts, has arrived at the following remarkable conclusion,
easily flxinff itself in the memory, and of great practical
value 1-^" that with imgumM, Iwfa la/rd (md olvoe oil, mi&r-
posed m aconOrmoue atraf/mn hetween them, surfaces of wood
on metal, wood on wood, metal on wood, amd meteil on metal
{■when in motifm), h<me aU of them very nearly the smne go-
efficdmtt of friction, the vakie <f thai co-efici&nt Idng in aU
cases mct/uded hetwe&n -07 and '08.

" J'br the imffuent taUoio, the co-effidemt is the same as for
the otliM wngumts m every case, except in that cf metals upon
metals. This imgueivt appea/rs, from the et^eriments of Mo-
ri/n, to ie leas suited to metaUio svhsto/nces thmi the others,
and gives for the m^eanvaiue of its co-effiaieni under the same
ci/rctmtstcmces '10."

134. "Whilst there is a remarkable uniformity in the results
thus obtained in respect to the friction of sun'aces, between
which a perfect separation is effected throughout their whole
extent by the interposition of a continuous stratum of the

* Thia result, of so much importance in tlie theory of mnehines, 'a fully esttt'
hliehed by the experimfintB of Morin.

f As, for iastance, with an oiicd or grfiiiaj clotli.

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FEICTION. 197

ungnent, there is an infinite variety in respect to those etatea
of imctuosity which occur between the emtr&ines, of which
we have spoken, of surfaces merely unctwoue* and the most
perfect state of lubrication attainable by the iuterpc^ition
of a given unguent. It is from this variety of states of the
uuctuosity of rubbing surfaces, that so great a discrepancy
haa been found in the experiments upon friction with ungu-
ents, a discrepancy which has not probably resulted so much
from a difference in the quantity of the ungnent supplied to
the rubbing surfaces in Qifl'erent experiments, as in a diffe-
rence of the relation of the insistent pressures to the extent
of rubbing surface. It is evident, that for every desciiption
of unguent there must coiTcspond a certain pressure per
square inch, under which pressure a perfect separation of
two surfeces is made by the inteiposition of a continuous
stratum of that unguent between tnem, and which pressm-e
per square inch being exceeded, that perfect separation can-
not be attained, however abundant may be the supply of the
unguent.

The inge/mous experiments of Mr, Kicholas "Woodf, con-
iiiTned by those of Mr. G, Iiennie$, have fully estabHshed
these important conditions of the friction of unctuous surfaces.
It is much to be regretted that we are in possession of no
experiments directed specially to the determination of that
particular pressure per square inch, which corresponds in
respect to each unguent to the state of perfect separation,
and to the detennmation of the co-efGcienta of fnctions in
those different states of separation which correspend to pres-
sures higher than this.

It is evident, that where the extent of the surface sustain-
ing a given pressure is so great as to make the pressure per
square inch upon that surface less than that wliich corres-
ponds to the state of perfect separation, this greater extent of
surface tends to increase the friction by reason of that adJie-
siveness of the unguent, dependent upon its greater or less
viscosity, whose effect is proportional to the extent of the
surfaces between which it is interposed. The experiments
of Mr. "Wood§ exhibit the effects of this adhesiveness in a
remarkable point of view.

* Or slightly cubbed with the unguent.

+ TreatisB on Enil-roacla, Srd ed, p. 399.

± Trans. Eoyal Soc 1S2B.

5 It ia evident thut, whilst by extending Ihe nnetnous surface which sustains
any giren pressure, we diininiBh the oo-efficient of ftietion up to a certain
Bmit, we at the same time increase that adhsdon of the surfaces which results

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138 FEionoN.

It is perhaps deserying of enqiiiry, whether in respect te
those considerable presBures under which the parts of tlie
larger machines ai'e accustomed to move tipon one another,
the adhesion of the unguent to the surfaces of contact, and
the opposition presented to their motion by its viscidity, are
causes whose influence may be altogether neglected as com-
pared with tlie ordinary friction. In the case of lighter
machinery, as for instance that of clocks and watches, these
considerations evidently rise into importance.

135. The experiments of M. Morin show the friction of
two enrfaces which have been for a considerable iim<i in con'
tad, to be not only different in iis amount from, the friction
of surfaces in oontwAioits "motion, but also, specially in tJm,
that the laws of friction (as stated above) are, in respect to
the friction of quiescence, subject to cames of variation and
imcertwmi/y from which the friction of motion is exempt.
This variation does not appear to depend upon the extent of
the snrfaces of contact, in which case it might he referred to
adhesion ; for with different pressures the co-efflcient of the
friction of quiescence was found, in certain cases, to vary
exceedin^y, altihough the surfaces of contact remained the
same.* lie uncertainty which would have been introduced
into eveiy question of construction by this consideration, is
removed by a second very important fact developed in the
couree of the same experiments. It is this, that by the
slightest jar or shoch ot two bodies in contact, their friction
is made to pa^ from that state which accompanies quiescence

from tlie viscosity of the viijgaent, so that there ma j be a point where the gain
on the one hand begins to be exceeded by the loss on iie other, and where
the surface of minimum rasistance under the given pressui-e is therefore
attained.

Mp. Wood eonmders the pressure pec aquare inch, wMoh corresponds to the
mimmura reastanoe, to be SOlbs. in the case of axles of wrought iron turning
upon cast iron, with fine neat's foot oiL The experiments of Mr. Wood, whilst
they place the gensral results stated above in full eyidenoe, can scarcely how-
ever be considered satisfactory as to the particular numerical values of the con-
etimts sought in this inquiry. In those experimenta, and in others of the same
class, the amount of friction is determined from the observed space or time
through which a body projected with a ^yen velocity moves before all its
velocity is destroyed, that is, before its accumulated work is eihausted. This
is an easy method of eiperimeat, but hible to many inaccuracip>i It is much
to be regretted that the experiments of Morin were not eitonded to the fric-
tion of unctuous snrfaces, reference being had to the pressure per square

* Thus In the caKe of oak upon oak with parallel fibres the coefficient o(
the friction of quiescence varied, uiidtr diilntnt [rps&urLJ upon the same Sur-
face, from 'SB to ■76.

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FRICTION. 139

to that wliicli accompanies motion ; and ae every machine or
structure, of wliatever kind, may be considered to he subject
to sucli shocks or imperceptible motions of its suifaces of
contact, it is evident that the state of friction to be made
the baeis on -which all questions of statics are to be deter-
mined, should be that wnich accompanies continuous motion.
The laws stated above have been shown, by the experiments
of Morin, to obtain, in respect to that friction which accom-
— '"s motion, with a precision and uniformity never before
id to them ; they have given to all our calculations in
^ t to the theory of machines (whose moving surfaces
have attained their proper hearings and been worn t-o their
natural polish) a new and unlooked-for certainty, and may
probably be ranked amongst the moat accurate and valuable
of the constants of practical science.

It is, however, to be obsei-ved, that aH these experiments
were made under comparatively small insistent pressures as
compared with the extent of the surface pressed (pressures
not exceeding from one to two kilogrammes per square cen-
timeter, or from about 14'3 to 28'6 lbs. per square inch.) In
adopting the results of M. Morin, it is of importance to bear
this fact in mind, because the experiments of Coulomb, and
particularly the excellent experiments of Mr. G. Eennie, car-
ried far beyond these limits of insistent pre^ure*, have fully
shown the co-efflcient of the friction of quiescence to increase
rapidly, from some limit attained long before the surfaces
abrade. In respect to some surfaces, as, for instance, wrought
iron upon wrought fron, the co-ef&cient nearly tripled itself
as the pressure advanced to the limits of abrasion. It is
greatly to be regretted that no experiments have yet been
directed to a determination of the precise limit about which
this change in the value of the co-effleient begins to take
place. It appeara, indeed, in the experiments of Mr, Ren-
nie in respect to some of the soft metals, as, for instance, tin
upon tin, and tin upon cast iron ; but in respect to the harder
inetals, his experiments passing at once from a pressure of
32 lbs. per square inch to a pressure of 1-66 cwt, per square
inch, and the co-efflcient (in the case of wrought iron for in-
stance) from about -148 to -25, the limiL which we seek is
lost in the intervening chasm. The experiment of Mr. Een-
nie have reference, liowever, only to the friction of qui-
escence. It seems probable that the co-efficient of the tric-

* Mr, Rennle^e espepimenta were carried^ in some cases, to froiu 5 cwt. t*>

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130 I-EICTIOS.

tion of motion remains constant under a wider range of pres-
sure than that of qiiiescence. It is moreover certain, that
the limits of preaera-e beyond which the surfacea of contact
begin to destroy one another or to abrade, are sooner readied
when one of tbem is in motion upon the other, than when
they are at rest: it is also certain that these limits arenot in-
iependent of the Telocity of the moving surface. The dis-
cussion of this subject, as it connects itself especially with
the friction oi 'moUon, is of great importance ; and it is to be
regretted, that, with the means bo mimiflcently placed at his
disposal by the French Government, M. Morin did not ex-
tend his experiments to higher pressures, and direct them
more particularly to the circumstances of presanre and velo-
city under which a deatruction of the mbljiue surfaces first
begins to ahow itself, and to the amount of the destruction
of surface or wear of the material which corresponds to tlie
same space traversed under different pressures and different
velocities. Any accurate observer who ahould direct hie
attention to these aubjects would greatly promote the inter-
ests of practical acience.

SmiMAKT OF THE LaW8 OF PrICTION.

136. !From what hae here been stated it results, that if P
repreaent the pei^pendicular or no'i-mal force by which one
body ia pressed upon the surface of another, F the friction of
the two surfaces, or the force, which being applied parallel
to their common sui'face of contact, would cauae one of them
to slip upon the surface of the other, and^ the co-ejjicient of
friction, then, in the case in which no unguent is interposed,
/"represents a constant quantity, and (Art. 133.)

F=/P .... (109);

a relation which obtains acewatdy in respect to the friction
of motion, and ajpprtxmnately in respect to the friction of

137. The same relation obtains, moreover, i)
unctuoua sui-facea when merely rubbed with the undent, or
whore the presence of the unguent haa no other influence
than to increase the smoothness of the surfaces of contact
without at all separating them from one another.

In unctuoua surfacea ^ptsr^iti^ lubricated, or between which

./Google

THE LIMnlNG AHGLE OF EE3I8TAN0E. lol

a sfcratam of unguent is partially interposed, the co-efficient
of friction/ J8 dependent for its amount upon the relation of
the ioaiatent pressure to the extent of the surface pressed,
or upon the pressure psr squs/re inoh of surface. This
amount, eoiTesponding to each pressure per scjuare inch in
respect to the different unguents used in machines, has not
yet been made the subject of satisfactory experiments.

The amount of the resistance I" opposed to the sliding of
the surfaces upon one another is, moreover, as well in this
case as in that of surfaces perfectly lubricated, influenced by
the adhesiveness of the unguent, and is therefore dependent
upon the extent of the adhering surface ; so that, if 8 repre-
sent the number of square units in this surface, and o. the
adherence of each square unit, then aS represents the whole
adherence opposed to the sliding of the surfaces, and

F=/P+aS (110);

P

where yis a function of the pressure per square unit ^,and

a is an exceedingly small factor dependent on the viscosity
of the unguent.

The LiMirma Angle of Hesistanoe.

"We shall, for the present, suppose the parte of a solid body
to cohere so firmly, as to be incapable of separation by the
action of any foi-ce which may be impressed upon them.
The limits within which this suposition is true wdl be dis-
closed hereafter.

It is not to this resistance that our present inquiiy has
reference, but to that which results from the friction of the
surface of bodies on one another, and especially to the dweo-
Hon of that resistance.

138. Am/ pressure ajjpUed to the surface of a.. .

soUd howy iy the intervention of another body moveaUe
•upon it, teUl he smtamed hy the resistance of the mnfa^im
<f contact, what&om' be its mrection, ^provided <mhf the an-
gU wHcK that di/reetion makes with the ■perprnMoylar to
the surfaces of contact do not exceed a certain angle called

the LtMmNG AUGLE OF EE8ISTANCE of thoSC StrKFAOKS.

./Google

.oi THE LIMITING AHGLTJ OF EEBISTAlfCE.

This is tme, however great thepresswe may he. Also, if
the inclmation of the^preaswe to the perpen(Mcular exceed
the Umdting angle of resistaaioe, then th/ls presswe wiU not
he sustained hy the resistance of the swrjaces of contact /
and thda is true, however smaU the^essure may he.

Let PQ represent the direction in whicli the aurfacea oi:'
two bodies are pressed together at Q, and let
p QA be a perpendictilar or norrnal to the sur-
« faces of contact at that point, then will the pres-
sure PQ be sustained by the resistance ot the
surfaces, however great it may be, provided its
direction lie within a certain given angle AQB,
called the limiting angle of r^ietance ; and it will not be sus-
tained, however small it may be, provided its direction lie
without that angle, Por let this pressure be represented by
PQ, and let it be resolved into two others AQ and EQ, of
which AQ is that by which it presses the surfaces together
perpendicularly, and KQ that by which it tends to cause
them to slide upon one another, if therefore the friction F
produced by the first of these pressures exceed the second
pressure RQ, then the one body will not be made to slip
upon the other by this pressure "PQ, however great it may
be ; but if the friction P, produced by the perpendicular
pressure AQ, he less than the preaanre EQ, then the one
body will be made to slip upon the other, however small PQ
may be. Let the pressure in the direction PQ be repre-
sented by P, and the angle AQP by fl, the perpendicular
pressure m AQ is then represented by P ec«. fl, and therefore
the iriction of the surfaces of contact by/P cos. fi, / repre-
senting the co-efficient of friction (Art. 1S6.). Moreover, the
resolved pressure in the direction EQ is represented by P
sin. 6. ±he pressure P will therefore be sustained by the
friction of the surfaces of contact or not, according as

P sin. S is less or greater than fB e<s. ^ ;
or, dividing both sides of this inequality by P cos. 3, ac-
cording as

tan. S is less or greater than /.

Let, now, the angle AQB equal that angle whose tangent is
f, and let it be represented by ^, so that tan. ip—f. Substi-
tuting this value off in the last inequality, it appeal's that
the pressure P will be sustained by the fnetion of the sur-
faces of contact or not, according as

./Google

THE TWO STATES BOEDKlilMG UPON MOTION, 133

tan. 3 is greater or leas than tan. ^,
that is, according as

S is lesB or greater than <p,
or according as

AQP is less or gi-eater than AQB,
Therefore, &c. [Q- e. d.]

The Cone of E.b8i8TANCB.

139. K the angle AQB be conceived to revolve about the
axis AQ, so that BQ may generate the surface of
a cone BQO, then this cone is called the cone of
; EBSiSTANCE I it is evident, that any pressure, how-
ever great, apphed to the surfaces of contact at
Q win be sustained by the resistance of the sur-
faces of contact, provided its du-ection be any
where within the sui-faee of this eone ; and that it will not
be sustained, however small it may be, if its direction lie any
where without it.

The Two States bordering upoh Motion.

140. If the direction of tlie pr^sui-e coincide with the sur-
face of the cone, it will be sustained by the friction of the
surfaces of contact, but the body to which it is- applied will
be upon the point of slipping upon the other. The state of
the equilibriam of this body is then said to be that border-
ing TJFON motion. K the pressure P admit of being applied
in any direction about the point Q, there are evi<fently an
infinity of such states of the equilibrium bordering upon mo-
tion, corresponding to all the possible positions of P on the
surface of the cone.

K the pressure P admit of being applied only in the same
plane, there are but two such states, corresponding to those
directions of P, which coincide with the two intersections of
this plane with the surface of the cone ; these are called the
superior and inferior states bordering upon motion. In the
case in which the direction of P is limited to the plane AQB,
BQ and CQ represent its directions corresponding to the

./Google

134 THE TWO STATES EOEDERING UPON MOTION.

two states "bordeiing on motion. Any direction of P within
the angle BQC corresponds to a state of ec[iiilibrinm ; any
direction, ■without this angle, to a state of motion.

141. Since, when the direction of the pressure P coincides
with the snrface of the eone of resistance, the ecLuilibrium is
in the state bordering upon motion ; it follows, conversely,
and for the same reasons, that this is the direction of the
pr^aure sustained by the surfaces of contact of two bodies
whenever the state of their equilibrium is that bordering upon
motion. This being, moreovei', the direction of the pressure
of the one body upon the other is manifestly the direction of
the reaistmiee opposed by the second body to the pressure of
the first at their surface of contact, for this single pressure
and this single resistance are forces in etLoilibrium, and there-
fore equal and opposite. All that has been said above of the
single pressiire and the single resistance sustained by two
surfaces of contact, is manifestly true of the remUcmt of any
number of such pressures, and of the resvUamt of any num-
ber of such resistances. Thus then it follows, that when amy
mmiher of preasv/resajiplied to a body mmeahh v^on another
which ia fixed-, are sustained iy the resistance of t/is swrfoMs
<^ contact of the two "bodies, and a/re m the state of egml/i£ri/wm
borderi/iig v/pon motion, then- the direction of the resultant of
these pressures coincides with the etirfaoe<fthe cone of resist-
ance-, as does that also of the remMoM of the resistances of the
d^erentpoints of the mMfaces of contact*, thai is, they are
hoth inclmed to the perpendicular to the simfaoes of contact
{at the point where thmj irvt&rseet i£),atan angle equal to the
limiting angle of resistanee.

" The propevtieB of the limiting angle of realBtanoe and tlie e- .- „
anoe, were first given bj the author of this work in a, paper publialied lo t.'ie
Cambridge F

'he propevtieB of the limiting angle of reaistanoe and the "^"i" o'
were first given bj the author of this work in a paper publialiec
ridge Philosophical Transaotiona, toI. i.

, Google

Table I.
J*Hc<iim of Flame Burfaces, wlwn ihsy have bf

Limiting

BLr/acoa In Contact.

Dlspcialtion of

atihle (It the

"rSi™.!

Angis of

ESPEEIUENT3OfM.M0RIK

parallel \

without )
unguent f

O'ea

31° 48'

ditto 1

rubbed with i
dry soap \

(i-a

28 45

perpendiou-j

without 1

28 a2

36 23

Oalr uDOn oak

lar \
ditto

unguent f
with water

O'TX

tlie flut

without )

0'43

23 16

surface of

unguent j

the other

Oak upon elm

parallel

ditto

0'33

20 49

r

ditto

ditto

0'63

34 31 ■■

Elm upon oak - \

ditto

tubbedwith )

0'41

22 IB

1

perpendion-

Jar

without 1
unguent f

0-57

29 41

Ash, flr, beech, acmcc- 1
tree, upon oah 1

paraEel

ditto I

0-53

91 66

the leather

flat

ditto

0-61

31 23

Tinned leatheruponoakJ

the leather"
length-

ditto

0-43

23 16

steeped in )

o-ia

38 19

sidewajs _

water j

Jreeeed "*" °'|

parallel ■

without 1
unguent f

0't4

SB 30

perpendicu- ]

lar ]

ditto

0-47

25 11

parallel

ditto

0-eo

23 34

Hemp matting upon oak ■

ditto

steeped in 1

Q-m

41 a

Hemp cords upon oak -

ditto

witJiout J

0-80

38 40

!

ditto

0'62

Bl 4B

Iron upon oak

ditto

steeped in !

0-66

33 2

Cast-irnn upon oak

ditto

ditto

0-65

83 2

Copper upon oak -

ditto

without

0-62

31 48

steeped in

0'B2

31 48

flat or side-

■rtT*'^'' ■!

upon cast-iron [ | ways

tallow, Qv

O'la

6 51

hog's lard 1

, Google

DisposiUop of

Slaleoftlis

Co.m.i.nt

Angieof

the Fibres.

Bui-faoes.

ofPriCici,

EipsriuentsofU, UORIN.

-continued.

Black dressed leather, or)

(

witliout }

strap leather, upou ft [

unguent f

cast-iron pullej )

J

steeped

0-33

20 49

Oast-iron upon Cast-iron -

ditto j

without )
uuguent f

0'16

9 a

Icon upon cast-iron

ditto

ditto

019

10 4B

Oak, elm, joke elm, iron,-]

j

witli tallow

5

cast-iron, and braes 1

witli oil, or 1

sliding two and two, f
one upon another

j

hog's iard y

O'lSj

8 32

Calcareoua oolite atone )

fiiHc i

without 1

O'H

30 30

upon calcareous oolite f

ditto j

unguent )

Hard calcareous stone,

eallec! muaohelitalk, ;

ditto

ditto

0-75

86 62

upon calcareous oolite

BHck upon calcareous 1
ooUte

ditto

ditto

0-Sl

33 60

Oak iipOQ calcareous

wood end- (

o

oolite

ways \

"

Iron upon calcareous oolite

flat

ditto

O'iS

26 1

Hardeakareou>i3tone,or

muacliell.alk, upon

ditto

ditto

o-io

35 0

muschelkalk

Calcareous oolite stone
upon mu^thelkalk

ditto

ditto

0-15

86 S2

Brick upon musth^lkalk -

ditto

ditto

O'flT

33 SO

Iron upon musclielkalk -

ditto

ditto

0-42

2a 4T

Oak upon muscbelkalk -

ditto

ditto

with a coat- "1

ingofmor-

0-64

82 38

Calcareous oolite stone 1
upon calcareous oolite (

ditto

tar,of three
p^rtsoffiue -

sand and
one part of

Black lime .

0-:4g

38 80

* The surfaces retaining a<

I When the contact has not lasted long enough to eipress the grease.

j: When the contact has lasted long enough to espreas the grease and briijg

g After a contact of from ten to iiitecn minutes.

./Google

K„.,...,...MO„.,..

S^S

Ilmitine
Anglo.

Soft caleareons stone, well dressed, upon the same

Oominon brick, ditto . - - - -
Oak, endways, ditto . . - - -

Wrought iron, ditto

Hard calcareous st^>iis, well dressed, upon hard calsare-

Soft, ditto

Common brick, ditto

Oak, endwajs, ditto

Wrought iron, ditto - - - -
Soft cslcareoua stone upon soft calcareous stone, with
fresh mortar of fine sand . , • .

EXFERIHENTS BI DlFFEKEIfl' OESEBTIRS.

Smooth free-stone upon smooth free-stone, dry (Rennie)

Ditto, with fresh mortar (Rennie)

Hard polished calcareous stone upon hard pollEhed cal-

Calcareous stone upon ditto, both surfaces being made

rough Tfith ft chisel (Bon«Uardi)
Well dressed granite upon rough granite (Rennie)
Ditto, with fresh mortar, ditto (Rennie) -
BoK of wood upon pavement (Regnier) -
Ditto upon henten earth (Herbert)
Uhage stone upon a bed of dry clay
Ditto, the clay being damp and soft -
Ditto, the day being equally damp, but cOTered with

thick sand (Grere) - - - , -

0'14

o-e:

0-63

o-io
o-';e

0-04
0-43

0-14
0-71

0-78
0-88
0'48
0-58
0-33
0'61
0-34

0-.10

U° 80'
SB 52
33 60
32 13
26 1

35 0

36 63
83 60
32 3T

22 41

36 30

35 28

30 1

37 68
S3 26
26 7
30 1
18 16
21 2
18 47

21 43

, Google

FHdion of Plane Surfast

B Motion one tipon the other.

LimiilllB

Surfices in CootBot,

Disporition of
the fibres.

SuiCacea.

ofFrloUon,

Angle ot

Experiments OP M.MoRiN.

parallel j

without )

0-4B

25=39'

ditto

rubhedwith [
dry soap (

0-lB

a 6

perpendicu-

without [

Oak upon onk

ditto

wood end-'

water f

0'25

14 3

length-

without )
unguent j

04.

10 48

ways

paiBllel

ditto

0-43

2S 11

Elm upon oak

pecpeudicu- j

ditto

0-43

24 14

parallel

ditto

0'23

14 3

Ash, fir, beeoli, wild pear-

0-36 to
0'40

1 19 48
fai 49

tree, ami seiriee-tree, ■
upon oak

ditto

ditto ]

ditto

0'6a

SI 48

14 35

Iron upon oak

ditto J

rubbedwith)
dry eoipf

0-ai

11 sa

without )
unguent f

6-M

28 1

with water

0-22

12 25

Cast-icou upon oak

ditto

rubbedwith

10 46

dry soap

Copper upon oak -

ditto ■

without
unguent

0'B2

81 48

Iron upon elm

ditto

ditto

0'25

14 3

Cast-iron upon elm ■

ditto

0'£0

11 19

Blaok dressed leather)

ditto

ditto

0-27

la 1

upon oalc f

Tanned leather upon oak-

length-
ff ays, and
edgeways _

ditto \
with water

B'30to
0-3B

16 42
19 18

IH 11

without )

0'56

29 16

Eteeped in i

0-36

19 48

Tanned leather upon i

ditto

water f

greased and 1

steeped in t

0-2S

12 58

with oil

0-16

8 32

, Google

Sucr^crs la Cout&ct.

DlEpoBlUon Of
Ihe ribrea.

Slat* of the
aur(ac«3.

Co Qfficiaiil

AllBle of

ExPERIMEHtS OF M. MoBlN.

Hemp, in threads or in

parallel i

without 1
unguent J

0-52

27029'

cord, upon oak 1

perpendiou.

Willi water

0-33

18 16

Oat and elm upon cast- i"

parallel

without )

unguent j
ditto

O'SS

20 49

Wild peai^tree, dilto -

ditto

0'44

2S 46

Iron upon iron

ditto

ditto

0-44

23 45«

Iron iipon cast-iron and 1

ditto

ditto

O-lSf

10 13

Cast-iron, ditto

ditto

ditto

0-15

8 32

( upon brass -

ditto

ditto

0-20

11 10

Brasa 4 upon cast-iron -

ditto

ditto

0-23

12 25

{upon iron - -

ditto

ditto
greased in'

O'liiJ

0 B

Oak, eto, yoke elm, mid-]

way with
tallow,

O'OTto

\t 35

pear, cast-icon, wrought
iron, steel, and moving V

ditto

hog's lard,

O-08§

one upoaajiotlier, or on

gom
slightly )

greasy to J-

O'lo

8 32

thi- touch )

Calcareous oolite stone )

witlioul

upon calcareous oolite t

unguent j

Cftloareoua stone, called i

ditto

0-67

83 60

careous oolite )

Common briok upon oal- 1

ditto

dittu

0'6S

33 2

Oak upon oaloaceouh (

wood end, 1

oolite j

ways J

Wionght iron, ditto

parallel

ditto

0-69

34 37

rdiu3ehelkall(,npon mus-

ditto

ditto

0-38

20 49

obelkalk

CaleareoiiB oolite stone
upon mnieheltilk

ditto

ditto

o-es

33 2

Common brick, ditto

ditto

ditij)

0-60

30 58

Oak upon muskliflkalk i

wood end-
^ays

ditto

0-38

20 49

ditto

0-24

Iron upon muacbelkilk -

parallel

saturated
1 with water

0-30

16 42

* The lurfacs w^ar when there is no ,
{ The surtaxes stdl retaining a little n
§ When the grease is constantly renewed and uniforn
proportion can be reduced to O'Oo,

, Google

I'riction of Ovdgmna or Axle-sntlt, in Motion, upon their Bearings.
(From the eiperimenta of Morln.)

Iron Ksles 1
lignnm vitce
bearings

Brass axles i

oated with oil of
olives, with hog's
lard, taliow, and

greasy anfl wetted

coated with oil of

olives, with liog's

lard, tallow, and

soft gom

greasy

grea^lOid damped
scarcely greasy
without unguent
with oil or hog's

lard

greasy with ditto

greasy, with a

mixture of hog's

lard and molyb-

coated with oil

of olives, tallow,

hog's lard, or

Goft gom
coated with oil of

oliTes,hog'slard,

or tallow
coated with hard

gom
greaay and wetted
scarcely greasy
coated with oil,

or bog's iard
greasy

coated with oil
with hog'e lard

j 4° 0'
' S 6

10 48
5 9

6 17
10 46

* The surfaces beginning t<

, Google

SnrfiiceB Id ContHct.

'"'-^t^'^lf/lt^^^X-

Uu>m^

la Uie usueJ

ContlaBouslj.

lignum Tito!
axles, ditto 1

axlee in lig- 1
num vilEe f

bearings J

coated with hog's

lard
greasy

coated with hog'fl
lard

0-15

!■ •

O'OT

60S1'
S 82

i 0

Table IV.

■ of Friction tinder Pressures increastsd ctintinuaUi/ tip t
lAmits of Abrasioit,
(From the eiperimentB of Mr, G, Eennie.'")

Co-effld^UptPrJotlon. ]

S,™™ iBQh.

ght-lron g^

^ upon Br

Bsupoa

Wcusht-lron Ca

OJ..

B2- Elba.

■140

174

166

157

1-66 owt.

260

376

soo

225

2'00

271

202

219

2-3S

285

321

340

214

2-66

297

329

344

211

8-00

312

i88

347

216

S-S3

350
3-6

S61
353

351
353

209

ao6

31b

208

4'33

8(15

866

S66

231

5-00

40J
4r9

3G6
367
367

357

358
859

223
238

3B7

fl'OO

876

403

288

B-33

434

234

6-66

235

7-00

282

1-S3

273

» Phil. Trans. 1829, table 8. p. 1

./Google

THE KJGIDITY OF COEDS,

THE RIGIDITY OF COEDS.

lis. It ia evident that, by reason of that i

deflexion which constitutes the ri-
gidity of a cord, a certain force or
pressure must be called into action
whenever it is made to change its
rectiUneal direction, so as to adapt
itself to the form of any curved sur-
face over which it is made to pass ;
as, for inatauce, over the circumfe-
rence of a pulley or wheel. Sup-
pose such a cord to sustain tensions represented by 'P, and
F„ of which P, is on the point of preponderating, and let
the friction of the axis of the pulley be, for the present,
neglected. It is manifest that, in order to supply the force
necessary to overcome the rigidity of the cord and to pro-
duce its deflection at B, the tension P, must exceed P, ;
whereas, if there were no ri^dity, P, would equal P, ; so
that the effect of the rigidity in increasing the tension r, ie
the same as though it had, by a certain quantity, increased
tile tension P,. Now, from a very numerous series of
experiments made by Coulomb upon this subject, it appears
that the quantity by which the tension P, may thus be con-
sidered to be increased by the rigidity, is partly constant
and partly dependent on the amount of P, ; so as to be
represented by an algebraical fonnula of two terms, one
of which ia a constant quantity, and the otlier the product
of a constant quantity by Pj. Thus if D represent tlie
constant part of this formula, and E the constant factor
of Pj, then is the effect of the nudity of the cord the same
as though the tension P, were mcreased by the quantity
D-i-E .T,.

When the cord, instead of being bent, under different
pressures, upon cu'cular arcs of equal radii, was bent upon
circular arcs of different radii, then this quantity D-l-E.P,;
by which the tension P, may be considered to he increased
by the rigidity, was found to vary inversely as the radii
of the arcs ; so that, on the vrhole, it may be represented
by the formula

./Google

THE RIGIDITY OF (

K

• (HI),

where R represent* the radius of the circular arc over which
the rope is bent. Thus it appeal's that the yielding tension
P, may he considered to have been increased by the rigidity
of the rope, when in the state bordering upon motion, so as
to become

Thia formula applies only to the bending of the same eord
under different tensions upon different circular arcs : for dif-
ferent cords, the constants D and E vair (within certain
limits to be specified) as the sgwwes of the diaineteTS or of the
circumferences of the cords, in respect to new corda, wet or
dry; m respect to old cords they vary nearly as thej>ower i
of the dia/mMers or drcmnferences.

Tables have been fm'nished by Coulomb of the values of
the constants I) and E. 'Kiese tables, reduced to English
measures, are given on the next page.*

• The rigidity of tb© cord exerts ita influence to increase reaistance only at
tliat point wliere the cord winds u[)on tlie pulley ; at tbe point where it leaves
the pulley its eiastidtj fayours ratJier, and does not perceptibly affect, the
conditloDS of the equilibrium.

In all calcnlatJons of machines, in whicb the moving power is applied by the
intervention of a rope passing over a pulley, one-ltaf Un diameter gf rope is
to he added to the radius of tite pitlleg, or to the psrpendic«lar on the <lireclion
of the rope from the point whmce the moments are taeatwred, tte pressure

./Google

! KlGinriT OF COEDS.

Table V. Hiqiditt of Ropes.

TabU of the values of the eamtimU D aii<J E, oceoi-dtrwj (o tlu experimeiUa <
Coulomb {ndiicei to UngUsh msamrei). The radius R of ilie pidley is to
taken iitfeet.

Value of D in lbs.

Value of E in lbs.

Ciroumfevenue of
the Hope in Intlios.

Value of Din lbs.

ValueofEinlljg.

a

4

■288053

i-oaaai';

I6-886a06

■OOS1S76
■0230303
■0731755
■3B34860

No. S. Dry half-worn ropes. lOgidity proportional to
of the cube of the circumference.

Circumference of
thn Rope in Indies.

ViiIueofDinlbs.

Value ofE ill lbs.

1
2

■149272
■41!ifl36
1 ■1606*1

S-S0B7a7

■0064033
■0180327
■05iail5
■144823S

No. 4. Wettea half-wora cords. Rigidity proporUonal
to the square root of the cube of ths di-cumfercnce.

';ir-

'—I

Circamferenca of
the Rope in Inches.

Value of D in lbs.

V«lueofEinlba.

i

■Bia

I

■2(12511
■627338
2'339B75
6-616589

■006401

■oism

•061212
■144822

, Google

THK BIGIDITY OF COKDS.
No. 5. Tarred rope. Rigidity proportional to tlic u

Numtier of Strauds.

ViLlue of D m lbs.

Vi,Iue of E \n lbs.

15
80

1'252&4

U-044983

To detemdiie the constants D and E for ropes wlioae circumferences are
Intewnediate to those of the tables, find the ratio of the given circumference
to that nearest to it in the toblea, and seek this ratio or proportion in tlie first
column of Uie amiliary table to the right of the page. Tho corresponding
number in the second column of tWs auxiliary table is a factor by which tbe
vslnes of D and E for the nearest circumference in tbe principal tables being
multiplied, their values for the given circumfereiioe will be determined.*

» Note (s) Ed. App.

./Google

THE THEOET OF

F A.R T III.

THE THEOET OF MACHINES.

ii3. The parts of a machine are divisible into those whieli
recevee the operation of the moving power immediately, those
which operate iimnediaiiJ/y upon me work to be performed,
and those which comjrmmcaie' ietween the two, or. which
conduct the power or work from the moving to tihe worhmg
points of the machine. The iirst class may be called keobiv-
EES, the second oPEitATOEs, and the third t
work.

The Teah8mi88ion of. "Work by MAcnnjEa.

144. The moving power divides itself whilst it operates in
a machine, first, Into that which overcomes the prejudicial
resistances of the machine, or those which are opposed by
friction and other causes -uaeleasly absorbing the work in its
transmission. Secondly, Into that which accelerates the
motion of the various moving pai-ts of the machine ; so long
as tlie work done by the moving power upon it exceeds that
expended upon the various resistances opposed to the motion
of the machine (Art. 139.). Thirdly, Into that which over-
comes the useful resistances, or those which, are opposed to
the motion of the machine at the working point or points
by the useful work which is to be done by it. Thus, then,
the work done by the moving power upon the mvoing -points
of the machine (as distingmshed from the worki/ng points)
divides itself in the act ol transmission, first, Into the work
I nselessly upon the friction and other prejudicial
opposed to its transmission. Secondly, Into that
2 in the various moving elements of liie machine,
and reproducihle. Thirdly, Into the useful work, or that
done by the operators, whence results immediately the useful
products of the machine.

./Google

THE THEOET OF MACHINES. 347

145. Thi aggregate nv/rtiber of units of ■meful works yielded
hy any maehine at its worhmg jmnis is less than the mmir-
ier received upon the Tnaehine dvreoily from, the moving
^loer, hy fhemimher of units ^^endediipon t} — ' ''

oial resistances cmdhythe mmb&r-of .wmts m _

in the moving parts of the m.aoh/metmiilst the work i

For, by the principle of vis viva (Art. 139.), if 2TJ, repre-
sent tlie nranber of unite of work reeeiyed tipon the machine
immediately from the operation of the moving power, lu
the whole numher of sticn units absorbed in overcoming the
pr^uddoial resistances opposed to the working of the ma-
chine, 2TJ, the whole usejw, work of the ]nachine (or that
done by its operators in producing the useful effect), and

~^w{p^—v^) one half the aggi'egate difference of the vires

vivje of the various moving parts of the machine at the
conamencement and teraiination of the period during which
the work is ^timated, then, by the principle of vis vrvA

■ (112) ;

in which v, and v^ represent the velocities at the commence-
ment and termination of the period, during which the work
is estimated, of that moving element of the machine whose
weight is w. Now one-hall the aggregate difference of the
vires vivEe of the moving elements represents tlie work aecu-
tmdated in them during the period m repect to which the
work is estimated (Art 130.). Therefore, &c.

146. 7^ the same velocity of mery part of the machine re-
turn after any period of ttm£, or if the motion he pmiodical,
then ie the whole work received upon it from the moving power
dmwig that time exactly eqiud to the sum of the useful work
done, and the worh emmded tmon the pre^udioid, resistances.
For ihe velocity bemg in this case the same at the com-
mencement and expiration of the period during which the
work is estimated, lw{v'~v,')=0, so that

* Note (*) E(J, App.

./Google

lis THE M0DUL08 OF A MACHINE.
2U,=2ir, + 2w (113).

Therefore, &c.

The converse of this proposition is evidently true.

liT. If the prime mover m a machine be throughout th6
motion vti equ^dbrium witib the useful and the pre^tiddeiai
resistances, tlien the motion of the machine w uniform.
For in this ease, hj the principle of virtual velocities
(Art. 127.), sU,=2U, + 2'w; therefore (equation 112)
Sv>{v^''—v,^)=0; whence it follows that (in flie case sup-
posed) tlie velocities «, and v, of any moving element of the
machine are the same at the commencement and termi-
nation of any period of the motion however email, or that
the motion of every such element is a uniform motion.
Therefore, &c.

The converse of this proposition is evidently true.

The Modulus of a iMACHiNE moyikg wrrn a unifoem or
, Motion.

-48, The modulus cf a m^aehme^ in the sense i/n which the
term is used in- this work, ia the relation between the work
constanih/ done v^on it hi/ the moving j^wer, and that con-
stantly yielded at the working points, when it has attained
a state of umiforvn -motion JT *^ admit of such a state of
motion ; or if the natu/re of its motion oe p&riod^eal, then
is its modulus the relation between the worh done at its
mi&m/ng and at its working poimis in the inierval of H/rns
which it ocGttpies in passing from amjgiv&n velodty to the

Tlie modulus is thus, in respect to any machine, the parti-
etilar form mpUccMe to that machine of equation (113), and
being dependent for its amount upon the amount of. work 2m
expended upon tlie friction and otlier prejudicial resistances
opposed to the motion of the various elements of the ma-
chine, it measures in respect to each such machine the loss
of work due to tliese causes, and therefore constitutes a true
standard /!>7' comparing the ^i^endUnre of Tnovvng power n&-
■f to the prodmstion of the same effects by different ma-

./Google

THE MODULUS OF A MACHINE. 149

chines: it is tlius a measure of the working quaUtJes of

"Whilst tlie particular modulns of every differently con-
Btnicted machine is thiis different, thera is neTertheless a
general algebraical type or formula to which the moduli of
machines are (for tlie most pai't and with certain modifica-
tions) referable. That form is the following,

TJ,=A.U,+B.S (114),

where tJ, is the work done at the moving point of the ma-
chine through the space 8, TJ, the work yielded at the work-
ing points, and A and B constants dependent for their value
upon the construction of the machine : that is to say, upon
the dimensions and the combinations of its parts, their
weights, and the co-efficients of friction at theh' various rub-
bing surfaces.

It would not be difficult to establish generally ^\w, form, of
the modulus under certain assumed conditions. As the mo-
dulus of each particular machine must however, in this work,
be discussed and determined independently, it will be better
to refer the reader to the particular moduli investigated in
the following pages. He will observe that they are for the
most part compnsed under the form above assumed ; sub-
ject to certain modifications which arise out of the discus-
sion of each individual case, and which are treated at length.

149. There is, however, one important exception to this
general form of the modulus : it occurs in the case of ma-
chines, some of whose parts move immersed in fluids. It is
only when the resistances opposed to the motion of the parts
of the machine upon one another are, like those of friction,
proportional to the pressures, or when they are constant re-
sistances, tiiat this form of the modulus obtains. If there be
resistances which, like tliose of fluids in which the moving
parts are immei-sed (the air, for instance), vary with the velo-
city of the motion, and these resistanceB be considerable,
tiien must other tenns be added to the modulus, Tliis sub-
ject will be furtlier discussed when the resistances of fluids
are treated of. It may here, however, be obseiTfed, that if
the machine move vrnforinhj subject to the resistance of a
fluid during a given time T, and the resistance of the fluid

./Google

he supposed to vary as the square of the velocity V, ther.
will the work expended on this resistance vary as V^ , S, or
as V . T, since 8=Y . T. If then TJ, and U, represent the
work done at the moving and working points during the
time T, then does the modulus (equation 114) assume, in this
case, the form

U,=A . U,+B . V . T + C . y= . T (llS).

The Modtilus of a Machine movikg with an .

I MOTIOX.

150, In the two last articles the work IT,, done upon tlie
inoving point or points of the machine, has been supposed to
be jtist that necessary to overcome the useful and prejudi-
cial resistances opposed to the motion of the machine, either
continually or periodically ; so that all the work may be ex-
pended upon these resistances, and none accumulated in the
moving parts of the machine as the work proceeds, or else
that the accumulated work may return to the same amoxmt
from period to period. Let us now suppose this equality to
cease, and the work U, done by the moving powei' to exceed
that necessary to overcome the useful and prejudicial resist-
ances ; and to distinguish the work represented by U, in the
one case from that m the other, let ns suppose the former
(that which is in excess of the resistances) to be represented
by U' ; also let U, be the useful work of the machine, done
through a given space S,, and which is supposed the same
whatever may be the velocity of the motion of the machine
whilst that space is being described ; moreover, let S, be the
space described by the moving point, whilst the space S, is
being described by the working point.

Now since IT, is the work which must be done at the
moving point just to overcome the resistances opposed to
the motion of that point, and IT' is the work actually done
upon that point by me power, therefore IT'— U, is tlie excess
of the work done by the power over that expended on the
resistances, and is therefore equal to the work aocumulated
in the machine (Art. 130.) ; that is, to one half of the
increase of tbe vis viva through the space S, (Aii. 129.) ; so
that, if V, i-epresent the velocity of any element of the
machine (whose weight is w) wlien the work IT' began to be
done, and v^ its velocity when that work has been com-
pleted, then (Art. 139.),

./Google

Now by equation (114) U,=AD",+BS„

.-. U'=:A . TJ,+B . S.+^Swiv.'^v,') (110).

If instead of the work TJ' done by the power exceeding that
U, expended on the resistances it had been less than it, then,
instead of work being accumulated continu^ly through the
space S„ it would continually have been lost, and we should
hare had the relation (Art. 129.),

[j.-U'=|»(«-,'-'>.');

80 that in this case, also,

The equation (116) applies therefore to the case of a
retarded motion of the machine as well as to that of an
accelerated motion, and is the general expression for the
modulus of a machine moving with a variable motion.
Whilst the co-efficients A and B of the modulus are depen-
dent wholly upon the friction and other direct resistances to
the motion ot the machine, the last tei-m of it is wholly
independent of all these vesietances, its amount being deter-
mined solely by the velocities of the variouB movmg ele-.
ments of lihe machines and their respective weights.

The Vklocity of a MAciiraE movixg with a vakiable
Motion.

151. The velocities of the difE'crent parts or elements of
every machine are evidently connected with one another by
certain invariable relations, capable of being expressed by
algebraical formulse, so that, altliough these relations are
din'erent for different machines, they are the same for all
circumstances of the motion of the same macliine. In a
great number of machine this relation is expressed by a
constant ratio. Let the constant ratio of the velocity v, of
any element to that V, of the moving point in such a

./Google

152 1

macliine be represented by >-, bo that v^=i>•Y^, and .et v, and
Vj be any other values of v, and V, ; then ti^^^XV^. Sub-
stituting these values of v^ and v^ ia equation (116), we
have

U'^A . U,+B . S. + ~(y,=-T.')2y.V (lit);

m which expression 2;/;V represents the eiim of the weights
of all the moving elements of the machine, each being mul-
tiplied by the square of the ratio ?. of its velocity to that of
the point where the machine receives the operation of its
moving power. For the same machine this co-efflcient 2wV
is therefore a constant quantity. For different machines it
is different. It is wholly independent of the useful or pre-
judicial resistances opposed to the motion of the machine,
and has its value deternuned solely by the weights and
dimensions of the moving masses, and the manner in which
fchCT are connected with one another in the machine.
Ti-ansforming this equation and reducing, we have

by which equation the velocity V, of the moving point of
the machine is determined, after a given amount of work
U' has been done upon it hy the movmg power, and a given
amount TJ, expended on the useful resistance ; the velocity
of the moving point, when this work began to he done
being given and represented by Y,.

It is evident that the motion of the machine is more
equable as' the quantity represented by 2mX.° is greater.
Tais quantity, which is the same for the same maclune and
different for different machines, and which distinguishes
machines fi'om one another in respect to the steadiness of
their motion, independently of all considerations aiising out
of the nature of the resistances useful or prejudicial opposed
to it, may with propriety be called tJie co-efficieht of
EQUABLE MOTION.* The actual motion of the machine is
more equable as this co-eiBcient and as the co-efficients A
and B (supposed positive) are greater.

s here, for the first time, introdueeiJ

, Google

CO-EFFICIENTS OF THE

To DETEEMINE THE Co-EFFICIENTS OF THE MoDtiLL'S OF A
KACHrtTE.

152. Let that relation first be deterinined between the
moving pressure Pi npon the machine and its working pres-
sure P„ which obtains in the state hordermgimon motion by
the preponderance of P,. This relation wiH, in aU cases
where the ooriMmd resistancea to the motion of the machine
independently of P, are small aa compared with P,, be
found to be represented by fonnxilse of which tiie following
is the general type or form : —

P,=P, .*,+*, (119);

where *, and *, represent certain fmietiony of the friction
and other prejudicial resistances in the machine, of which
the latter disappears when the resistances vanish and the
former does not ; so that if */"> and *,("^ represent the
values of these functions when the prejudicial resistances
vanish, then *,f'"=0 and $,(">= a given finite quantity
dependent for ite amount on the composition of the machine.
Let Pj^"' represent that value of tlie pressure P, which would
be in equilibrium with the given pressure P^, if there were
no prejudicial resistances opposed to the motion of the
machine. Then, by the last equation, P,(^'=P, . */">.

But by the principle of virtual velocities (Art. 137.), if
we suppose the motion of the machine to be -wniform, so
that P, and P, are constantly in equilibrium upon it, and if
we represent by Si any space described by the point of
application of P„ or the projeoHon of that space on the
direction of P, (Art. 52.), and by S, the corresponding
space or projection of the space described by P,, then
P,W . Si=P, . S,. Therefore, dividing tliis equation by
the last, we have

S.= §i, (120).

Multiplying this equation by equation (119),

P..S,=V,.S..|^,|+S.{i^i|=P..S,|A_|+s_.,,.

•••"D.= !Ai(.U,+»,.S (121)

, Google

15-1 AXES,

which is the modulna of the machine, so that the constant

A in equation (114) is represented by r-^,, and the constant

B by ^,.

The abOTe equation haB been proved for any value of S,,
provided the values of P, and P, be constant, and the
motion of the machine uniform ; it evidently obtains, there-
fore, for an exceedingly smaU value of 8„ when the motion
of the machine is vanahle.

G-ENE]iAL CoNDrnOX OF THE StATE BOBDUBING UPON IIOTION

ns A Body acted upon bt Prkssuees ih the same Plane,

AND MOVEABLE ABOUT A CYLmDRIGAL AxiS.

153, ff any ■mtmhm- of pressures P„ P^, Pj, <&o. OfppUed in
the same plans to a hotly moveable about a cyrnidrical
aids, he in t/te state bordering upon motion, then is the
d/i/reotion of the resistance of the oasis indlmed to its radius,
at the jioint where it irtterseots the avrowir^erence, at an
(mgle eqital to the lindting angle of resisia/noe.

For let R represent the resultant of P, P„ &c. Tlien,
since these forces are snpposed to be upon the
■ point of causii^ the axis of the body to turn
upon its bearings, their resultant would, if made
to replace tliem, be also on the point of causing
the axis to tm-n on its bearijigs. Hence it fot
lows that the direction of this resultant R caimot
I be through the centre C of the axis ; for if it
V were, tlien the axis would be pressed by it in the
direction of a radius, tliat is, petpendioularly
upon its bearings, and could not be made to turn npon tliem
by that preasnre, or to be upon the poiiit of tnniing upon
tliem. The direction of R must then be on one side of C,
so as to press the axis upon its bearings in a direction liL,
inoUned to the normal CL (at the point L, where it inter-
sects the circumference of the axis) at a certain aiiglo ELC.
Moreover, it is evident (Art. 141.), that since this force R
pressing tlie axis npon its bearings at L is upon the point of
causing it to sUp u])on them, this inclination ItLC of R to
the perpendicular OL is eqnal to the limiting angle of

./Google

THE WHEEL AND AXLE.

155

resistance of the axis and its bearings.* Now tlie r
of the axis is evidently equal and opposite to the resultant
li of all the foreee P„ Pj, &e. impressed upon tlie body.
This resistance acts, therefore, in the direction LR, and is
inclined to CL at an angle equal to the limiting angle of
resistance. Tlierefore, &c.

The "Wheel ahc Axle.

154. The pressv/res P, aml'P.a^^
UcalJy oy means of pa/rallel cords to a
wheel amd csde are m the state hordering
'wpon motion hf the prefpondeTwnce (f P^
^ ii fegv/ired to dete/rmime a rdaUon
between F, amd P,.

The direction LK of the resistance of the axis is on that
side of the centre which ia towards P„ and is inchned to the
perpendicular CL at the point L, where it intersects the
axis at an angle CLE equal to the limiting angle of resist-
ance. Let this angle be represented by <p, and the radios
CL of the axis by p ; also the radius CA of the wheel by a^,
and that OB of the axle by a, ; and let "W be the weight of
the wheel and axle, whose centre of gravity is supposed to
be C. Now, the pressures P„ Pj, the weight vf of the
wheel and axle, and the resistance E of the axis, are pros'
snres ia equilibrium. Therefore, by the principle of the
equality of moments (Art. 7.), neglecting the rigidity of the
cord, and observing that the weight "W" may be supposed to
act through 0, we have,

P. .OA=P, .CB + R.^.
If, instead of P, preponderating, it had been on the point
of yielding, or P, nad been in the act of prepondei'ating,
then E would have fallen on the oUier side of C, and we
should have obtained the relation P, . CA=P5 . OB —
E . C^; so that, generally, P, . CA=P. . CB±R . C^;
the sign ± being taken according aa P, is in the swperior or
inferior state bordering wpon motion.
Now CA=a„ CB=a„ ^=CL sin. CLR=p ein. 9, and

" Tlie xiile of C on which EL falls Is manifest!)' determined by the direction
towards whieh the motion ia about to tHlie place. In this cnse it is aiippoacd
obont to take place to the rijAi of C. [f it had liecn lo the Ufl, the direc-
tion of R would liu.vfi been on the opposite side of C.

, Google

5 WHEEL ATfD AXLE.

'R=P,+P,± W; tlie sign ± being taken according aa &e
weight W of the wheel and axle acta in the same direction
with the preaaurea P, and P„ or in the opposite direction ;
that is, according aa the pressures P, and i*, act vertically
dmonwards (as shown in the figure) or upwards /

.■.Pa=Pa+(P.+P,±'W) p sin. 9,

.■.P,((j!,— p sin. <)))=:P,(a,+p sin. ip)±Wp sin. ip.

Now the effect (Art. 143.) of the rigidity of the cord BP,

is the same as though it increased the tension upon that cord

from Pj to lPj+ — ■ ■ "1 : allowing, therefore, for the

rigidity of the cord, we have finally

PjH — - — ^1 (t^j + p ein. ip)±"W p sin. <p,

or reducing,

EWtPBHi^ k r . , (122),

which is the required relation hetween P, and Pj in the
state bordering upon motion,

— sin. (p and — sin. f are in all cases exceedingly small ;

we may therefore omit, without materially affecting the
result, all terms involving powers of these quantities above
the first, we shall thus obtain by reduction

(l.l4),.„.,}..a.B,

155. TM modulus of v/rviform motion, in the wheel <md axle.

It is evident from equation (122), that, in the case of the
wheel and axle, the relation assumed in equation (119)

./Google

I> + (^±W)p8m.?

Kow observing that *,*"' represents the value of *, when
the prejudicial resistances vanish (or when 9=0 and E=0),

Therefore l)y equation (121),

^AB"

^-&'

D+ (2±wjf sin.?

!

— p sm. ^

■ (12ft),

which is the moduhis of the wheel and axle.

Omitting teiixis involving dimensions of — sin, c

P E

— sin, a. and — above the first, we have

(i:+i±5)<'™-''}--(i^=')-

156. y/te modulus of variable motion in the wheel and OAtle.

K the relation of P, and P, be not that of either state
bordering upon motion, then the motion will be continually
accelerated or continually retarded, and work will continu-
ally accumvilate in the moving parts of the machine, or the
work already accumulated there will continually expend

./Google

THE WHEEL t

itself until the whole is exhausted, and the machine ie
brought to rest. The general expresaion for the modulna in
tliia state of variable motion is (equation 116)

this case of the wheel and axle, if V, and V, re-
present the velocities of P, at the
commencement and completion
of the space S„ and a the angulai-
velocity of the revolution ot the
wheel and axle ; if, moreover, the
preesures P, and P, be supposed
to be supphed by weights sus-
pended from the cords ; then,
since the velocity of P, is repre-

sented by -^— -, we have iwv,"^^

PX'+P,(?^')"+«>.I,+.V,I.,if

1, repiesent the moment of inertia of the revolving wheel,
and X that of the levolvmg axle, (Art. 16.), and if jj-, repre-
sent liie weight ot a imit of the wheel and f^^ of the axle ;
since iwWi' repi esents the sum of the weights of all the mov-
mg elements of the machine, each being multiplied by the
BCLaare of its velocity, and that (by Art. 75.) iVJ, represents
this sum in respect to the wheel, and aV,Ij in respect to the
axle. Now, Y,=aa^,

y.^ jF,a,'+P.<+l',I, + l'J.l

j P,< + P,»," + t',I,+»,1, 1 .

Similarly 2yWj'=V5'

.■.^a(v;-v;)=(V;-V,'j

;p,<+p.<+i'.i.+tfa

Substituting in the general expression (equation 116), ■
have

./Google

THE WHEEL AND AXLE. 15S

U'=AU, + BS.+ilT,'-T,')
|E*!±2£^+iii±A} ...(126),

which is the modulus of the machine in the state of varialjle
motion, the co-effieienta A and B being those already deter-
mined (eCLTiation 124), whilst the co-efficient

FA°+FA°+>^.I,+t*A ia tije co-efficient iw^.' (equation

a, ^

117) of egwMemoUon. Ifthe wheel and axle be each of them
a solid cylinder, and the thickness of the wheel be i„ and the
length of the axle 5„ then (Art. 85.) \=^\a\, X=^'b^a^.
Now if "W, and "Wj represent tlie weights of the wheel and
axle respectively; then ~W ^^lea^b^^^, 'W ^■='^a'l}^^ ; therefore
(*Ji=i" it^i'? V-X=^'^^^^' Therefore the eo-efflcient of
ecLuable motion is represented by the equation

2wX'=P.+iW. + (P,+il\^)n^' (137).

157. To deiermme the mhcity acqvdred through a gw&n
apaoe 'when thereloMonofthe weights V^ and V^, suspended
from a wheel and axle, is not that of the state lordemig
v^on, 'motion?'

Let Si be the space through which the weight P, moves
■whilst its velocity passes from V, to V, : observing that

TJ'ssPiS,, and that U^=P,Ss=Pj-i--',aubatitntinginecLua-

tion (126), and solving it in respect to Vj, we have

• Note (m) Ed. App.

./Google

jnaMng the same suppositions as in formula 127, and repre-
senting the ratio — by m-, we have

f2^S,

158. If the radius of the axle be taken equal to that of the
■wheel, the wheel and axle becomes a pul-
ley. Assuming then in equation 122,
a^=a,z=a, we obtain for the relation of the
moving pressures P, and P„ in the state
bordering upon motion in the pulley, when,
the strings are parallel.

1— -8iu.9
and by equation 124 for the

J) + \-±^jfBin.v
a—? sin. 9

of the modulus,

(130);

.(129);

in which the sign ± is to be taken according as Uie pressures
P, and P, act downwards, as in the first pulley of the pre-
ceding figure ; or upwards, as in the second. Omitting

dimension of - sin ip, - sin.

by equations (123, 125)

p^pil + S+^yiilj.

e the first, we have

-g4)'

psin. 9 [..(131)

./Google

BT8TEM or OHE FIXED ONE

Also observing tliat a,=a^, and I,=0, the modulus of varia-
ble motion (eq^iiation 136) becomea

U'=AU,+BS-|-i(V,'-V,')jP,+P,-|-i:"WJ (133),

and the velocity of variable motion {equations 118, 128) is
determined by the equation

T'=T.- + 2i,s{|^|i^| (134);

ID which two last equations the values of A and B are those
of the modulus of equable motion (equation 125).

System of oni; Frxi® and one moveaele Pulley.

159, In the last article (equation 131) it v
ebown that the relation between the tensi*
Pj and Pj upon the two parts of a string pass-
ing ovei- a pulley and pai-allel to one another,
was, in the state bordering upon motion by the
preponderance of P„ represented by an expres-
sion of the form P,=^a.P,-(-5, where ts and 6 are

r] constants dependent upon the dimensions of the
pulley and its axis, its weight, and the rigidity
of the cord, and determined in terms of these
elements by equation 131 ; and in which ex-
j pression h baa a different value according as the
tension upon tiie cord passing qver any pulley
;ame direction with the weight of that pulley (as
in the iiret pulley of the system shown in tlie figiire), or in
the opposite direction (as in the second pulley) : let these
different values of S be represented by J and J,. Now it is
evident that before the weight P, can be raised by means of
a system such as that shown in the figui'e, composed of one
fixed and one moveable pulley, the state of the equilibrium
of both pulleys must be that bordering upon motion, which
is described in the preceding article ; since both must be
upon the point of turning upon their axes before the weight
T% can begin to be raised. If then T and * represent the
tensions upon the two pai-ts of the string which pass round
the moveable pulley, we have

./Google

162 BTSTKII OF ONE FIXED AXD ANT

l\-aT+h, and T=at + 'b,.
Now the tensions T and i together support the weight P„
and also the weight of the moveable pulley,

Adding aT to both sides of the second of the above equa-
tions, and mtdtiplying both sides by a, we have

Also multiplying the firet ec[uation by (1 + a),
{\ + a-}P,=a{l+a)T+l{l + a)=a\V,+y^)+al, + l{l+a),

,P,=.(^)p,+«:MM±^. . . .(135).
Now if there were no friction or rigidity, a would evi-
dently become 1 (see equation 131), and 't>°= -..-— - would
become-; the co-ef&cients of the modulus (Art. 148.) are

\l-\-ar 1 + a '

... U.=a(^)u.+f:^±fkt2)±*g (13C),

which is the modulus of unifonn motion to the single move-
able pulley,*

If this system of two pulleys had been
an-anged thus, with a different string passing
over each, instead of with a single string, as
shown in the preceding figure, then, represent-
ing by t the tension upon the second part of
the string to ■which P, is attached, and by T
that Tipon the first part of the string to which
Pj is attached, we have

P,=a^-f-6, T=»P, + 5, P.-|-i-i-"W=T.

" Tho modQluB may be determined directly f cj t n (1S6); for It ta

evident that if Sj and Si represent the epacea d ib d h anie time by

P, andPa, then Si — aSi. Multiplying both d f q t (ISf ) by thie
equation, ive liave,

iw>

TT„ therefore &e.

, Google

NCMBER OF

Multiplying the last of these equations by a, and adduig it
to the fli-st, we have P,(l + a)+Va~Tffl+S=(iT, + (l+a)5;

^M-^)

P, + i-

. (13?),

. (138).

and for tlie modulus (equation 121),

It is evident that, since the co-efficient of the second term
of the modulus of this syeten is less than that of the first
system (equation 136) (the quantities a and h heine essen-
tially positive), a given amoimt of workvU, may he done by
a less expense of power U„ or a gived weight P, may be
raised to a given height with less worh, by means of this
system than the other ; an advantage which is noi due
entirely to the circumstance that the weight of the move-
able pulley in this case acts in favour of the power, whereas
in the other it acts agamst it ; and which advantage would
exist, in a lees degree, were the pulleys without weight.

A System of one fixed and any iNrMBEE <:
Pulleys.

' M0VEji3LB

160. Let there be a system of n moveable
pulleys and one fixed pulley combined as
shown in the figure, a separate string passing
over each moveable pulley ; and let the ten-
sions on the two parte of the string which
passes over the firet moveable ptiUey be re-
presented by T, and t^. those upon the two
parts of the string which passes over the
second by T, and t„ &g. Also, to simplify
the calculation, let all the pulleys be sup-
posed of equal dimensions and weights, and
the cords of eqnal rigidity ;

.'.eliminating, T,=

.■.T,=a«.4-&„ andT,+W=T, + ^.;

. (139).

Wa + 6,

Let the
and^;

o-efficients of this equation be represented by r

./Google

I OF ONl^ FIXED 1

Similarly, T,=«T, + ,8, T,=aT,+/3, T,= tiT.+^, &c.=&c..

Multiplying these eq^uations successively, beginning from
the aecona, by », «■', a', &c., t"-*, adding tnem together, and

striking out terms (
equation, we have

or summing the geometrical progret
member,

i of the resulting

. (140);

Substituting for a. and /3 their values from equation (139),
and reducing

lfciwP,=iiT,+S;

Whence observing, that, were there no friction, a wotild

become unity, andl^j 1 =\-A . "We have (equation 121)

for the modulus of this system,

161. If each cord, instead of having one of
its extremities attached to a fixed obstacle, had
been connected by one extremity to a move-
able bar carrying the weight P, to be raised
(an arrangement which is shown in the second
figure), then, adopting the same notation aa
before, we have

T,=ff*,+5, (ti,+&=T„ T,-T,+(,+'W.

Adding these equations together, sti-iking out
terms common to both sides, and solving in
respect to T„ we have

\aVU

a+1

|W;

./Google

KUMBEE OF MOVEABLE TCLLETS. 165

in whicli equation it is to be observed, tliat tlie symbol h
does not appear ; that element of tlie resistance (wliicli is
constant), affecting the teneione t, and ^5 equally, and there

fore elimicating with Tj and T^. Let -
by «, then

- be represented

t,=<it--'W. Similarly, )!,=<— -W,
i,=a^-^,&c.=&e.,i„_i='^„-^'W

. {143).

Eliminating between these equations precisely as between
the similar equations in the preceding case (equation 140),
observing only that here /3 is represented by — gW, and that
the equations (143) are n—1 in number intead of n, we have

^.--x-^r-^) (i«)

Also adding the preceding equations (143) together, we have

t,+t,+ . . . +^„-i = «(#,+;>+ . . - t,)-{n-l)-!L

Now the pressure P, is sustained by the tensions i„ t„ &c.
of the different sti-ing^ attached to the bar whicli carries it
Including P„ therefore, the weight of the bar, we have

and t,+ . . . +i„=P,— i^^;

.•.P,-i^=K(P,-;,)_(n-l)^.

:.t={l-o)-2,+«.tM^-'^)--^
Substituting this value of #, in equation (144),

i,=(l-a)tt''-iP, + c."i^ + (ra
Transposing and reducing,

(i-.-)i,=(i-.K''P.+^

, Google

106 TA.OKLE OF ANY NCJIIiER OF BHEA.VEB.

■ P -^ hWJ

-•)•-!

. (146).

Whence obsei-vmg that when (S=l, Kl + a ^)" — 11=2" — 1^
we obtain for the modulus of unitorin motion (equation
181),

.TJ.+

■ ((l + o-')-

A Tackle op ast Number of Sheaves.

162. If an number of pulleys (called m tliis ease sheaTes)
be made to turn on as many different centres in tbe same
block A, and if in another block B there be sirni.
larly placed as many others, the diameter of e.
of the last being one half that of a con-espc
ing pulley or sheave in the first ; and if the s:
cord attached to the first block be made to pa^
in succession over all the sheaves in the two
blocks, as shown in the figure, it is evident that
the parts of this cord 1, 2, 3, &c. passing between
the two blocks, and as many in number as there
are sheaves, v?ill be parallel to each other, and
will divide between them the pressure of a weight
P, suspended from the lower block : moreover,
that they would divide this pressure between
them equaUy were it not for the friction of the
'» sheaves upon their bearings and the rigidity of

the rope ; so that in uiia case, if there were n sheaves, the

tension upon each would be -P, ; and a' pressui'c P, of that

,y Google

ANY NOMBEE OF SHEAVES. 167

amount applied to the extremity ot" the cord would be suffi-
cient tomamtainthe equilibriuin of the state h ordering upon
motion. Let T„ T„ %, &c. represent tlie actual tensions
npon the string in the state bordering on motion by the pre-
ponderance of F„ beginning from that which passes from P,
over the largest sheai ; then

&c.=&c., T^,=a.T„+5„ ;

where a„ a„ &c., h„ &„ ifec. represent certain constant co-
efficients, dependent upon the chmensions of the sheaves and
the rigidity of the rope, and determined by equation (131).
Moreover, since the weight P, is supported by the parallel
tensions of the different strings, we have

P,=T,-fT,+. . . . -|-T„.

It will be observed that the above equations are one more
in number than the quantities T„ T„ T„ &c. ; tlie latter may
therefore be eliminated among them, and we shall thus ob-
tain a relation between the weight P, to be raised and that
Pj necessary to raise it, and from thence the
modulus of the system.

To simplify the calculation, and to adapt
it to that form of the tackle which is com-
monly in use, let us suppose another ar-
rangement of the sheaves. Instead of their
being of different diameters and placed all
in tiie same plane, as shown in the last
figure, let them be of equal diameter and
placed side by side, as in the accompanying
figm-e, which represents the common tackle.
The inconvenience of this last mode of ar-
rangement is, that the cord has to pass from
the plane of a sheaf in one block to the plane
I of the corresponding sheaf in the other ob-
) Uquel/y, so that the parts of tiie cords be-
tween the blocks are not truly parallel to
one another, and the sum of their tensions is not tmly equal
to the weight P^ to be raised, but somewhat greater than it.
So long, however, as the blocks are not very near to one an-
other, this deflection of the cord is inconsiderable, and the
error resulting from it in the calculation may be neglected.
Supposing the different parts of the cord between the blocks
tlien to be parallel, and the diameters of all the sheaves and

./Google

163 TACKI.I5 OF ANY NUMBFJi OF HHEAVES.

tlieir axea to be equal, also neglecting the influence < f tlio
weight of each sheaf in increasing the friction of its axis,
since *]iese weights are in this case comparatively small, the
co-efflcients d,, «„ a, will maniteatly all be equal ; as also
h, K h ;

&c.=&c., T_,=t(T„+& f ^■^*'''

also P,=T,+T,+T,+ +T„.

Multiplying eq^iiations (liT) successively (beginning from the
second) by a, a", a'-, and a"-^ ; then adding them together,
striking out the terms common to both sides, and summing
the geometric aeries in the second member (aa in ecLuation
140), we have

Adding ec[uatioua (147), and observing that T,+T,+
.... +T,=iP„ and that P,+T,+T,+ .... +T„_i=
P, + P,— T„, wehave

P, + P,-T,=aP,+w5.
Eliminating T^ between this equation and the laat,

_._^ «^ ^«« &

tu"— 1 " a"— 1 a— 1 ^ ■'

To determine the modulus let it be observed, that, neglect-
ing friction and i-igidity, a becomes unity ; and that for this

value of a, — — ~- becomes a vanishine fraction, whose

value is determined by a well kaown method to be -*.
Hence (Art. 152.),

* Dividing numerator and denominator of the friioljon by (a— 1) it becomcE

-^-y^ q;; -j^, wMoh eTidently equals - when 0=1. Tlie roodulua

may readily be determined from eiiuadon (148). Let Si and Ss represent the
Bpacea described by Pi and Pj in any the same lime ; then, aoce when the
blocliS are made to approach one anotbet by the distance S-, each of the n por-
tions of the coed intercepted between the two blocks is aliortencd by this dis-

, Google

THE MODULUS OF A COMPOUND JIAOHISE. 169

TJ _,,<^%^+ i !^^ 1^ Is. {149).

Hitherto no account bus "been taken of tlie work expended
in raising the rope which ascends with the ascending weight.
The correction is, however, readily made. By Art. 60. it
appeal's that the work expended in raising this rope {diffe-
rent parts of which are raised diiferent heights) is precisely
the same as though the whole quantity thus raised had been
raised at one lift through a height equal to that through
which its centre of gravity is actually raised. Now the cord
raised is that which may he conceived to lie between two
positions of P^ distant from one another by the space S„ so
that its whole length is represented by nS, ; and if /* repre-
sent the weight of each foot of it, its whole weight is repre-
sented by f-nSi, : also its centre of gravity is evidently raised
between the first and second positions of P, by the distance
^1 ; so that the whole work expended in raising it is repre-
sented by imSj' or by ^ — -, since S,=nSj. Adding this

work expended in raising the rope to that which would be
necessary to raise the weight P,, if the rope were without
weight, we obtain*

U.=«?1"41tT,+ i^^-^l .B,+ i^. .s- . . . (160),
s of the tackle.

The Modtjlus of a coMPOtnsD Machote.

163. Let the worh of a machine be transmitted from one
to another of a series of moving elements forming a com-
pound macliine, until from the moving it reaches the worlcmg
point of that machine. Let P he the jwesswe under which
the work is done upon the moving point, or upon the first
moving element of the machine ; P, that under which it is

tance S,, it is evident that the wliole length of cord intercepted between the
two bloelfS is ehortaned by nSj ; hut the whole of this cord roust have passed
over the first slieaf, therefore Si^bSj. Multiplymg equation (148) hy this
eqnalaon, and obaerying that Ui=PiBi and Uii=FjSi, we otatam the niodnlua
aa gJTon aboTS.

" A correction for the weight of the rope niaj be similarly applied to the
modulus of each of the otiier eystems of pulleys. The effect of the iodgltt of
the rape in increadag the eipendituro of work on tlie yWctioii of tlie pijle js if
neglected ^s unimportant to the lesnlt.

, Google

170 MODULES OF A COMPOCHD MACHINE.

yielded fvom the first to tlie second element of the machine ;
r, from the second to the tliird element, &c. ; and P^ the
preeaure under which it is yielded by the last element upon
the -useful pi-oduct, or at the working point of the machine.
Then, since each element of the compound machine is a sim-
^h machine, the relation between the pressures applied to
that element when in the state bordering on motion will be
found to present itself under the fonn of equation (119)
(Art. 152), in all cases where the pressm'e under which the
work upon each element is done is ^eat as compared with
the weight of that element (see Art. 166.).

Representing, therefore, by Kj, a„ a, . . . 5,, &„ &5 . . ., cer-
tain constants, which are given in terms of the forms and
dimensions of the severjd elements and the prejudicial resist-
ances, we have

&c.=&c., P^i=a,P„ + 5„.

Eliminating the n—1 quantities Pj, Pj, P, . . ., P^j, between
theee n equations, we obtain an equafion, of the form,

P=«P,+5 (151);

where a^a^a,a, . . . a , and

h=a^a^ . . . ci^J>.+ ci,a, . . .a^„i+ . • • + afi^+b,.

(132).

If tlie only prejudicial resistance to which each element is
subjected be conceived to be friction, and the limiting angle
of resistance in respect to each be represented by ip ; then
considering each of the quantities a„ o„ a„ i„ as a fanctio]i
of Ip, expanding each by Maclanrin's theorem into a series
ascending by powers of that variable, and neglecting tenns
which involve powers of it above the first, we nave

where, <!,<"', 5,*"', «/">, 5,'°', represent the values of ff„ 5„

o„ lEc, when (p=0 and \-t^\ , \~j^] ■> ^^

the similar values of tlieir first differential co-efficients.

./Google

(tY%=»<...,.,

(P=5W.^^, &c.=

Therefore a,= <) (!+«,), 5,= &.(">(1 + ^.). «-.= «.^"Hl+^),
J,=5,W(H-/3,), &c.=&c.; where «„ ^„ «„ ^„ &c., each
JiiTolving the factor ip, are exceedingly small. Substitutmg
the values of «,, «„ &c. in the expression for a, and neglect-
ing terms wliich involve dimensions of ^„ «j, &c. above the
first, we have

«=ffi/") aj.o'i .

) \l+'^,+ \^%+ ■■■■^\\ ....(153).

i&Tow the co-e£&cient of the firat teim of the modulus is

represented (equation 121) by — -^ a representing the co-

eiBcient of tlie first term of equation (119), also substitnting
the value of a from equation (153), and obsei'ving that

«(0)=aW.fflW «„TO, wehave-^= jl +«+... +«„!;

.•.U=!l+«,-|-»,-|-a.-|- .... -|-ajU„+5.S .... (t54),
which is the modulus of a compound machine of n elements,
U representing the work done at the mooing point, U,
that at the loorhing point, S the space described by the
moving point, and h a constant determined by equation
(182).

164. The cohditions of the equilibrium of aht two pees-
shres p, and p, afplied ih" the bathe plane to a bodt

MOVEABLE ABO0T A SIXED AXIS OF GIVEN DIMENSIONS.

In^^. 1. the pressure P, and P, are shown acting on oppo-
site sides of the axis

i hose centre is C, and
vafig. 2. upon the same
side. Let the direc-
tion of the resultant
ot P, and Pj be repre-
sented, in the first
case, by IR, and in
the second by Kl. It

./Google

173 AXE3.

ia in the dii'GC lions of these lines that the axis is, in the two
cases, pressed upon its bearings. Suppose the relation
between P, and P, to be such that the body ia, in both
cases, upon the point of taming in the direction in which
P, acts. This relation obtaining between P, and P^, it ia
evident that, if these pressures were replaced by their re-
sultant, that resultant would also be upon tlie point of caus-
ing the body to turn in the direction of P,. The direction
IE of the resultant, thus acting alone upon the body, lies,
therefore, in the first case, upon the same side of the centre
0 of the axis as P, does, and in the second case it lies .upon
the opposite side ;* and in both cases, it is inclined to the
radius QK at the point K, where it intersects the axis at an
angle OKE, equal to the limitinff angle of resistance {see
Art. 153.), Now, the resistance ot the axiB acts evidently in
both cases in a direction opposite to the resultant of P, and
P,, and is equal to it ; let it be represented by E. Upon
the directions of P„ P„ and E, let fall the perpendiculars
CA.„ CA,, and CL, and let them be represented by a„ a„
and K "fiien, by the principle of the equality of moments,
since Pj, P„ and E ai-e pressiu-es in equilibrium,

.■.Pa=Pa+>-e.

If P, had been upon the point of yielding, or P, on the
point of preponderating, then E would have had its direction
(in both cases) on the other side of C ; eo that the last equa-
tion would have h

P,(I, + '^E=Pa-
According, therefore, as P, is in the superior or inferior
state bordering upon motion,

Pa-Pa^(±M^-

And if we assume >v to be taken with the sign 4- or — , ac-
cording as P, is about to preponderate or to yield, then
qenercdUi

P,».~Pa=>-R (^55).

Now, since the resistance of the axis is equal to the resultant
of P, and P„ if we represent the angle PJP, by .f, we have
(Art. 13.)

* The arrows in the figure represent, not the dlreetions of the remltaaU
but of the resietanc^ of the aiis, which are opposite to the resultants.

■j- Care must be taJien to meaaare this angle, so th.it Pi and Ps may hare

./Google

Ii= yp,'^+2r,p, COS. j+iV-

Substituting this value of E in the preceding equation, and
ariiiaring both sides,

{Pa-Pa)'=^^XP.'+2P,P, cos. i + P,') ;
transposing and dividing by F/,

(pi) V."->--)-2(|)(»A+'''<!™-0= -«->-•>;

P, _ faoJ^+X' COS. i) ± -t/^aA+^' t^os- ')'— K' — ^') K'—^T.

. Pi_((?,a,+x' COS. i) J: ?■ |/(a|'+2a,q

"P, a.'-

Now let the radins OK of the axis be represented by p,
and the limiting angle of resistance OKR by ip ; therefore
x=OL=CK sin. OKK=p sin. (p. Also draw a straight line
from A, to A, in both figures, and let it be represented by L ;
.".«,'— 2(e,»5 COS. AjCA,+ffl,''=L'. Kow, eiace the angles at
A, and A, are right angles, therefore the angles A,IA, and
A,CA, are together equal to two right angles, or A,OAj+i
=w; therefore A,CAj=w— i, and cos. AjOA5=— cos. i;
therefore Jj'^aj' + 2a,a, cos. i+a,': substituting these values
of L° and >. in the preceding equation,

__(f](i«;+F' COS. isin.'ip) + p sin. <? (L'— p'sin. 'i sin. V)'
'~ (ffi,'— p'sin. >)

. P, . . (156).

The two roots of the above equation are given by positive
and negative values of X, they correspond therefore (equa-
tion 155) to the two states bordering upon motion. These
two values of X ai*e, moreover, mven by positive and nega-
tive values of <p ; assuming ther^ore 9 to be taken positively
or negatively, according as P, preponderates or yields, we
may replace the ambiguous by the positive sign. The

their dicectJons both tmoards or both frma the angular point I (as shown in the
figuve), and not one of them tomards that point and the other from it. Thus,
in the second figure, the inclination i of the preasucea Pi and Pa is not the
angle AjlP,, but the angle PjIPi. It la of importance to observe this distino
tion (see note p. IVE.).

./Google

^.=©

relation above determined between P, and F, evidently
satieties the conditions of equation (119). We obtain there-
fore for tlie inodAdv^ (equation 121)

(ffii(i^+ p° COS. I sin, 'p) + p(L'— p° sin, 'i sin", 'e/f sin, y
(a;'— p'sin. >)
V,... (15T).

If terms involving powers of ( — 1 sin. 9 above the fii"st be

neglected, that quantity being in all cases exceedingly
small, we have -j

f.= i©+©-'-}^- '^'''^'

D,= il + (J^\,i,,4u, (169).

165. To determine the resultant R of any number of pres-
sures P„ Pj, P, . . . ., m terms of those pressures, and the
cosmes of their inclinations to one (mother.

Let "■„ %, «„ &c. represent the inclinations
lAO, IBO, &c. of the several pressures Pj,
P„ &c. to any given axis CA in the same
plane; and let ii,, ',„ '„, &c. represent the
inclinations of these pressures severally to one
another,
Kow Z AIB= /IBO- ZIAC (Eue. I. 32.) ;

.', (,j=aj— ttj, /.cos, j„=cos. ct, COS. ttj+sin. \ sin, a^.
Similarly, cos, i„=cos. a, cos. cij+sin. a, sin. «,;,

COS. 1„=:C0B. Cj COS. a^ + SlQ. % siu. %.

Kow E'=(P, COS. «,+P. COS. s+P, COS. ^+ . . . )' + (P.
sin. a,+P, sin. a-^+V, sin. a^+ , . . )°, (equation 9, Art. 11.).

Squaring tlie two terms in the second member, adding the
results, and observing that cos. °«, + 8irL "i^i^l,
■R'=P/4-P,''+Ps'- . . +2P,P,(cos.aiCOS.«,+sin.a,sin.a,)
+ 2P,P, (cos. a, cos. ^+sin. », sin. «,)+ ;

./Google

.•.R''«=P,''+P;+IV+ . . . +3P,P,coa.i„ + 2P,P,<
l-2P,P,eoa.i„+ &c (160).

[66. Tee condixions op the equilibkium of three peks
9tike8, p,, i'j, p,, in thk same flaue applted to a b0d1
moveable about a fixed axis, the dieecnon of ose ob
them, p„ passing thkol'tth the centke op the axis, ane
the system being in the state bokdeking upon motion
by the pkeponderance of p..

„ represent the inclinations of the directions of
the pressures P„ P,, P, to one
another, a, and a^ the perpen-
diculars let fall from the cen-
tre of the axis upon P, and P„
and X the perpendicular let
fall from the same point upon
the resultant H of F„ P„ P,.
Then, since It is equal and
opposite to the resistance of
the axis (Art, 15R;), we have,
by the principle of the equality of moments, rfl^—^s^=
>>.It, for Pj passes through the centre of the axis, and its
moment about that point therefore vanishes.

Substituting the value of Efroin equation (160),

-PA=>^)Pi"+Pii''+P.' + 2P,P, COS. .,,+

2P,P, COS. i„ + 2P,P, COS. <,,.}'

ition, and transposing,

'.)! =

Pa-

Squaring both sides of this e

P_'(a^=_V)-2P, jP#,aj,-h>^'(P, cos.ii,+P, COS. i
-PX-^V {P,'+P/ + 2P,P, cos. '.si-
If this quadratic equation be solved in respect to Pj, and

* In whidi espreaaion it U to be understood that the inclination ii5 of tho
direetiona of any two forces is taken on the snppoaitlon that both the forces
act/rom or both act towarde the point in which they intersect,
ana not one toviaTda and the other /rom that point ; so that in
the case repreaenteil hi the accompanying figiire, the inclina-
tion I,, of the two forces Pi and Pj represented by the arroi
is not the angle PiIPi, bnt the angle QIPi, ainee IQ and IP, s
direelions of these two forces, both tending from thdr point
of iotetseotion i whilst the directiona of Pal and IPi
of them towards that point, and the other /rojn. it.

at DOin tne lorcea
ect, .

^ K

Jint ^ \

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terms -which nivolve powers of ^ ahove the first be omitted,
we shall obtain the equation

J (as in Art. 16i.} the line which joins the
feet of tlie pei'pendiculaj's, «, and a, by L, and tlie function
a, (a, COS. ',s + a, cos, i„) by M, and substituting for X its
value p sin. ip,

P.=:^^')P,+ -^) SP;P + P>.= + 2P,P,MJ** . . . (161).

Representing (as in Art. 153.) the value of P, when the
prejudicial resistances vanish, or when ip=0, by Pi'"', we

haveP,<'''=l — IP,. Also by the principle of virtual velo-
cities P,<''' . 8]=P, . g,. Eliminating ?,<"> between these
equations, we haveS,= I — 1 S,. Kultiplying equation (161) by

this, p,s =PA + -^-^^^^-^ ip;s,=p + 2P,p,s;Ji+] vs,vi *.

J 1 1 3 5 ^^^^ (SI ass .51,

Substituting U, for P,S„ Uj for P^S,, and observing that

p;sx 1 ^ (162.)

which is the modulus of the system.

If Pj he so small as compared with P, that in the expan-
sion of the binomial radical (equation 161), terms involving

P
powers of -p^ above the first may be neglected ; then,

" It will be shown in the appendix, that this eqnation ia but a particular
w of a more general relation, embracing the conditions of tlie equillbrinm
ujiy mimbsr of pressures applied to a body moTeable abont n cylindrical
ia of given dimenaona.

, Google

which eq^iation may he placed under the form

Whence oheerving that the direction of P, heing always
through the centre of the axis, the point of application of
Hiat force does not move, so that the force Pj does not work
as the body is made to revolve by the preponderance of P, ;
oheerving, moreover, that in this case the conditions of
equation (119) (Art. 153.) are satisfied, we obtain for the
modulus

fl.,..!,, ,/«■

^t= 1 + i^™- »["'?.+

(~.) (x)P.-S,- sill. »...■ (164)-

16T. The conditions of the eqmlih'iwm of two pressures P,
<md P, wppUed to a body moveable about a cylindrical axis,
taking into aeoount the w&ight of the hody and supposing i6
to le sym'metrical aiout its m:ia.

. The body being gymmetrieal about its axis, its centre of
gravity is in the centre of its axis, and its weight pi-oduces
the same efiect as though it acted continually through the
centre of its axis. In equation (161.) let tlien P^ be taken to
represent the weight w of the body, and i,^, i,, the inclina-
tions of the presaurea P^ and Pj to the vortical. Then

P,= (ajp,+ (L!H>^) j P,-I' + 2P,WM+WV I * . . (165.)

Also by the equation (162) we find for the modulus

\ a,a^ I ( »!

+"\v=s,v I * ■ ■ ■ (1^^-)

And in the case in -which P, is considerable ae compared
with "W, by ecLuations (163, 164).

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THE DIRECTION Of TIIK

U.^ jl + -^sin.9 iu,+ (^)wS, sin. ^., . (168.)

HJ8. A MACHINE TO WIIIOH AKE APPLIED AST TWO I

P, AND P„ AND WIIIOH IS MOVEABLE ABOUT A CTLINDEIOAL
AXIS, 18 WOKKEn 'WITH THE GREATEST ECONOMY OF POWER
WHEX THE DIEECTIOXS OF THK PRESSHKES AEE PAKALLEL,
AND "WHEN THET ARE APPLIED ON THE SAME SHIE OF THK
AXIS, IF THE WEIGHT OP THE MACHINE ITSELF BE SO SMALL
THAT ITS nJFLUEMCE IN INOEEABING THE FRICTION MAT BE
NEGLECTED.

For, representing the weight of such a machine by "W, and
neglecting terms involving W sin. 9, it appears by e(iuation
{168) that the modulus is

whence it foUows that the work U„ which must be done at

the moving point to yield a given amount XJ, at the working

point, is less as L is less.

N^ow L represents
the distance AjA, be-
tween the feet of the
perpendiculars OA, and
CA„ which distance is
evidently least when P,
and P, act on the same
aide of the axis, as in
fiff. 2, and when CA,

and CA5 ai-c in the same straight line ; that is, when P, and

F, are parallel.

169. A MACHINE TO WHICH ARE APPLIED TWO GIVEN PRES-
SDEEB Pi AND P, AND "WHICH IS MOVEABLE ABOUT A CYLIN-
DRICAL AXIS, IS WORKED WITH THE GREATEST ECONOMY OF
POWEK, THE DJFLrESCE OF THE WEIGHT OF THE MACHINE
BEING TATTTC^; INTO THE ACCOUNT, WTn::N THE TWO PRESSURES

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ECONOMY OF POWEE,

THE SiME BIDE OF TnE AXIS, AXD WIIEN
THE DlEEOTIuN OF THE MOVING- PEE8SDEE Pj IS ISCLTNED TO
THE VEBTK'AL AT A CERTAIN ANGLE WHICH MAY BE DETF.K-

Let P, be taten to represent the weight of the machine,
and let its centre of gravity coincide with the centre of its
axis, then is its modnltis represented {equation 166.) by

in which expression the wort Ui, which must be done at the
moving point to yield a given amoimt TJj of work at the
working point, is shown to be greater than that which must
have been done upon the machine to yield the same amount
of work if there had been tw) fricUon by the q^uantity

P am. 9

U,'L-+3U,P,S,M p? +P,'S.^

I i

The machine is worked then with the greatest economy of
power to yield a given amount of work, Uj, when Uiis func-
tion is a m,invmu7n. Substituting for L' its value

«,''+2a,ffjC0S. '„+«,', and for M its value a, {a, cos. i„ +

a, COB, !„} (see Art. 166.), also for Sj— J its value S,. it be-
comes

«,cos.>,,)+P,''Sx}*. ■ • -(169.)

Now let us suppose that the perpendicular distance a, from
the centre of the axis at which the work is done, and the in-
clination ij, of its direction to the vertical, are both giv&n, as
also the space S, through which it is done, so that the work
is given in every respect ; let also the perpendicular distance
a, at which the power is applied, and, therefore, the space S,
though which it is done, be gwen y and let it be required to
detei-mine that inclination i„ of the power to the work which
will under these circumstances give to tlie above function its
minimum value, and which is flierefore consistent with the
most economical working of the machine.

" " " 5" all the terms in tlie function (169.) which con-

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TEE DIRECTION (

lain (on. tlie above suppositions) only constant quantities, and
1 epresenting tlieir sum by 0, it becomes

^-^~ 1 ^''A^^t^' °'^^- '»+PA COS. >,,) + G \ *

"Now 0 being essentially posiiwe, this quantity is a mini-
mum when 2a,aJJJJJ, cos. i^+P^S^cos. i,j is a minimum ; or,
observing that Uj^P^Sj and dividing by the constant factor
2a,tJ,U,S,, -when

Trom the centre of the axia C let lines Gpi
Cp, he drawn parallel to the directions of the
pressures PiP, respectively ; and whilst C»j
and Cpj retain their positions, let the angle
^jOP, or i„ he conceived to increase until r,
attains a position in wliich the condition
P,cos. ii^ + PjCOS. ii3=a minimum is satisfied.
Jjfow p,CF^=p,Cp,—j>,GP„ or ',5=1,5—135 ;
' substituting which value of ii, this condition

becomes

P, cos. i,5+P3COS.{i,,— '„) a minimum,

or P, COS. lu+PjCOS. i„cos. i„+P, sin. i^sin. i,, a minimum,

or (Pj+Pj cos. '55) COS. i,j+P,8in. i,jsin. i„ a minimum.

_._(^P^_(.p^cos. i„)cos. ii,+(P,+P, COS. ij,) tan. 7 sin. i„ is a mi-
nimum, or dividing by the constant quantity (Pj+Pj cos. i^,)
and multiplying by cos. y,

cos. '„cos.7 + sin. ijjSin. 7=cos. (i,5— y) is a minimum.

"... (no.)

.-.,,,-.i-L«.i. I p^+p_ ,,,. ,

To satisfy the condition of a minimum, tlie angle p^Gp^
must tlierefore he increased until it exceeds 180° by that

angle 7, whose tangent is represented hyp- -°p '^^^ ^ . To

detennine tlie actual direction of P, produce then ^,C to q,
make the angle qCr equal to 7 ; and draw Cm- perpendicular

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GREATEST ECOXOMY OF POWi':K. IS"!

to Cr, aJid eqaai to the given perpendicular distance o, of
tiie direction of P, from the centre of the axis. If mP, be
then drawn tliroTigh the point m parallel to Gr, it will be in
the required direction ot P, ; bo that being applied in this
direction, the moving pressure Pi will work me machine with
a greater economy (S power than when applied in any other
direction round the axie.

it is evident that since the value of the angle 'isOrp^Op,,
which signifies tlie condition of the greatest economy of
power, or of the least resistance, is essentially greater than
two right angles, Pj and P, must, to satisfy that ooBDmoM,

BOTH BE APPLIED ON THE SAME SIDE OF THE AXIS. It is th&Th

a condUi'jn necesBO/ry to the most economical working of any
machine {whenever may ie its weight) which is mawahle dhovi
a ffylind/rical ams under Iajoo given presswes, that the mov-
ing- PEESSmiE SHOtTLD BE APPLIED ON THAT SIDE OP THE AXIS
OP THE MACHINE ON WHICH THE EESISTANCE IS OVEECOME, OB

niE WORK DONE. It is d furthef oondiUon of the greatest
econowy of power in suoh a 'machine, thai the di/vection in
which the moving pressure is applied ahovld he mdmed to
tJis vertical at an angle i,„ whose ta/ngent is detervrmied hy
equation (170.).

"When i5j=0, or when the work is done in a vertical
direction, tan. y=-0 ; therefore '1,=*, whence it follows that
the moving power also must in this case be applied in a ver-
tical direction and on the same side of the axis as the work.

"When J„=^ or when the work is done hoiizontally, tan.
1*=

-p ,

The moving power must, therefore, in this case, be applied
on the same side of the axis as the work, and at an incli-
nation to the horizon whose tangent equals the fraction
obtained by dividing the weight of the machine by the
working pressure.

3*
Since the angle ij, is greater than ■^ and less than -5-

eos. '„ is negative ; and, for a like reason, cos. 1,5 is also in
certain cases negative. Whence it is apparent that the
function (169.) admits of a minimum value under certain

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THE PULLIir.

conditions, not only in respect to the infUnation of the
moving pressure, but hi respect to the distance a, of its
direction from the centre of the axis. K we suppose tiie
space S, through which the power acts whilst me given
amount of worn U, is done to be given, and sabstitute in
tliat fanction for the product SjCt, its value S,«„ and then
assume the differential of the function in renpect to a, to
vanish, we shall obtain by reduction

TJ,;+2U,P,8,<

■.,+P='S,"

U,= co8..„ + U,P,S,<

■ (in.)

If we proceed in like manner assuming the space S^

of §1 to be constant and substituting in the function (169.)

for Sjffl, its value Sa*,, we shall obtain by reduction

__ F,a,

*'~ P,C08.I„ + P,C0S.V

It is easily seen that if when the values of i,, and i,, deter-
mmed by equation (170.) are substituted in these equations,
the resulting values of tSi are jiositme, they correspond in
the two cases to minimum values of the function (169.), and
determine completely the conditioim of the greatest economy
of power in the macbine, in I'espeet to the direction of the
moving pressure applied to it.

170, The pullet, wHiiN the tjiksions upon the two

EXTEEMITHLS OF THE COED HAVE XOT VEKTICAL UlEliO'l'IOiSS.

In the case in which the two pai-ts of the
string which pass over a pulley are not
pai-aflel to one another, the relations estab-
lished iu Article 158. no longer obtain ;
and we must have recourse to equation
(167.) to establish a relation between the
tensions upon them in the state bordering
upon motion. Calling W the weight of
the pulley, a its radius, and observing
that the etfect of the rigidity of the cord,
in increasing the tension P„ is the same
as though it caused the tension P^ to be-

come P, f 1 + — ) + — (Ai-t. 142.), we have

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THE PDLLET.

= (i+|){i + !i„i„.,|p.+

/L MW\ . 1 „„„,

where L represents the chord AB of the arc embraced hy
the string, and M=(('(oos. i„+cob. i,,), i„ and i„ represent-
ing the inclinations of Pi and P, to the vertical: which
inclinations are measured by the angles PiEP, and PjFP„
or their supplements, according as the corresponding pres-
sures P, and P, act downwards, as shown in the figure, or
npwards (see note to Article 165.); so that if both these

Eressures act upwards: then the cosines of both the angles
ecome negative, and the value of M becomes negative ;
whilst if one only acta upwai'ds, then one term only of the
value of M becomes negative.

Substituting this value for K, observing that L=2a cos. i,
where 3i represents the inclination of the two parts of the
cord to one another (so that 21=1,5+1,5), and omitting terms
which involve products of two of the exceedingly small

.-DE , p .
quantities —1 —, and -em. <? we have

'1 a a

si^s:^ ( ^' • ■ <" >•

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THE PULLET.

■which last equation is the modulus to the pulley, when the
t'wo parts of the sti'iug are inclined to the vertical and to
one another.

171. If both the strings be inclined at equal angles to the
Tei'tical, on opposite sides of it; or if i„=:ij,=:(, so tliat cos,
1,,+eoB I, =2 COB then equations (172.) and (173.) become

172. Il b tl J. \ita of tlie cord passing over a pulley be in
the same horizontal straight line, so that the
pulle} subtams no pressure resulting from the
tension upon the cord, but only bears its

tlien =:^, and the tenn involving

COS. t in each ol the above equations vanishes. It is, how-
ever, to be observed that the w&ight bearing upon the axis
of the pulley is m this cise the weight of the pulley
increased by the weight ot cord whicii it is made to support.
So that it the length of coid supported by the pulley be
represented by s, and the weight of each foot of cord by \i;
then is the weight sustained by the axis of the pulley repre-
sented by W-|-[ia. Substituting this value for "W m equa-
tion (175.), and assuming cos. '=0, we have

Cr,=(l + ^)u,+M D + CW-f-f^-s) p sin, 9 j S, (176.)

178. Let us now suppose that there are n equal pulleys
sustaining each tlie same length
s of cord, and let IT„ represent
the work yielded by the rope
(through the space S,) after it
has passed over the «.**, or last
pulley of the system, U, being
that done upon it before it
passes over tlie iirat pulley:
then by Art. 163., equations

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THE rrLLEY. 183

152. 15i. aiid 170., neglecting terms involving powers of

— , — , - sill. 0 above the iiret, and observing that Q;,=a,=
a a a ^

p sin. tp [■ , we have

U,= (l + «.|)u,+ ^JD + (W+(is)psin.9ls,.

Representing the whole weight of the cord sustained by the
pulleys by w, and observing that ^7is=w, we have

tJ.= (l+^)"[^.+^ I «-B+(ji.W+«')p sm. ? I S, . . . (17T.)

In the above equations it has been supposed, that altliongh
the direction of the rope on either side of each pulley is so
nearly hoiizontal that cos. i may be considered = 0, yet that
it does so far iend itself over each pulley as to cause the
surface of the rope to adapt itself to the circumference of
the pnUey, and thereby to produce the whole of that resist-
ance which is due to the -ngidAiiy oi the cord. If the tension
were so great as to cause the cord to rest upon the pulley
only as a rigid rod or bar would, then must we assume E^=0
and D=:0 in the preceding equations.

174. If one part of the cord passing over a pulley have a
horizontal, and the other a vertical direction, as, for mstanee,
when it passes into the shaft of a mine over tlie sheaf or
I wheel which overhangs its mouth ; then one of the

gles i,j or i„ (equation 173.) becomes xt ^^^ tlie

other 0 or t, according as the tension on the ve^
Heal cord is downwards or upwards, so that cos.
i,j+cos. 's5=±l, the sign i being taken according
as the tension upon the vertical cord is downwards
or npwai-ds. Moreover, in tliis case (Art. 170.)

I— -and c

• (ITS);

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THE PULLET.

U,= ll+?-+^siii.9lTI, + UD±^sm.9|s, . . (179)

174. The modifhts of a system of a/mj nwrJ>eT ofpuU&ys, vom

(WIS of vshieh the rope j
horizontally

'S v<irttcaUy, a

Let Ui repre-
sent the work
done upon tlie
Tope through
tlie space S, be-
fore it i^asaes
horizontally
over die first
^=. pulley of the
""^ system, and let
it pass horizon-
tally OT r «- such pulleys and then, after liaving passed
over anothei pulley of diflLient dimensions, let it take a
vertical direction descendmL foi instance, into a shaft. Let
Uj be the work yielded by it through the space 8, immedi-
ately that it has assumed this vertical direction : also let w,
represent the work done upon it in the horizontal direction
iniraediately iefore it passed over this last pulley of the
system. Then, by equation (179.),

E p 1^2 .

1u,4-Md.

Also, by equation (177.) representing the radius of each of
the pulleys which cawy tlie rope horizontally by o, the radius
of its axis by p„ and its weight by "W",, and observing that
Ui is here the jwwer and u, the work, we have

U,=(l + — W-f-ljnD + («"W,-l-w)p, Bin. v[s,.

Eliminating the value of m, between these equations; and
neglecting powei-s above the first in~, &c., we have

pf2 ,

"S^'-+!W^<)^

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(nW,+w)?,}

. (180.)

|i 175. If tlie strings be parallel, and Uieir common

a inclination to the vertical be represented by i, so

f tliat j„ = jj, =;i; then, since in tlus case L=2a, we

have (equation lT2j, neglecting tenns of more tlian

one dimension in — and L

1+^+-

. (181.)

in which equation i is to be taken greate

, . (182.)

■ or 1^8 than -, and

therefore the sign of cos. i is to be taken (as before explained)
positively or^ negatively, according as the tensions on the
cords act downwards or upwards. If the tensions are verti-
cal, 1=0 or *, according as they act apwat'ds or downwards,
so that COS. 1 = ± 1. The above equations agree in tliis case,
as they ought with equations (131.) and (132.). If the pai--

allel tensions are hwisontah then i=-, and the tenns involv-

' 3'

ing COS. 1 in the above equations vanish.

tl. iHE FiaCTIOK or A PIVOT.

When an axis rests upon' its bearings,
not by its convex circumference, but by
its extremity, as shown in the accompany-
ing figure, it is called a pivot. Let W
represent the pressure home by such a
pivot supposed to act in a direction per-
pendicular to its surface, and to press

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ISS THE rivoT.

equally upon every pai't of it ; also let pi represent the
radius of the pivot ; then will -^f' represent the area of the

pivot, and — - tlie pressure sustained by each unit of tliat

area. And if/' represent the co-efdcient of friction (Art.

133.), —^ will represent the force which must be applied

garallel to the surface of the pivot to overcome
-^.„^, 1® friction of each such unit. Now let the dot-
^''-^=^^^—-^ ted hnes iu the accompanying figure repreflent
an exceedingly narrow ring of the area of the pivot, and let
p and p+ip represent the extreme radii of this ring; then
will its area be represented by *(p + Ap)' — *p°, or by «■ i2p{Ap) +
{^f'f} , or, since Ap is exeeednigly email as compared with p,
by SiTpAp. Now the friction upon each unit ot this area ia

I'epresented by —^ ; therefore the whole friction tipon the

ring is represented by — ^ . S^pip, or by

*Pi r.

m&iiwnt of that friction about the centre of the pivot by

— J- . p'Apj and the sums of the momenta of the frictions of

Pi

all such rings composing the whole area of the pivot by

SWf 2Wf SW/ /•

2 — f~ . p^Ap, or by ^1— i^sp'Ap, or by —-— I pVp, or by

^i^\ or by iW/p, (183.);

whence it appears that thefriation of the pi/oot prod/uoes the
same effect to oppose the revolidion (f the mass which rests
iipon it, as though the whole pressti/re which ii sustains were
coUeoted over apovnt dista/niby two-thirds of its raddvsfrom
its centre.

If fl represent the angle through which tlie pivot is made
to revolve, then |-p,fl wul represent the space described by
the point last spoien of ; bo that the ^oor% expended upon
the reeietance nf acting there, would be represented by
^Wp,yS, which therefore represents the work expended upon
die friction of the pivot, whilst it revolves tJu'ough the angle

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; 80 that the work expended on each complete revolutiou
if the pivot is represented by

A^P^'W „ (184).

177, If the pivot he hollow, or its surface be an annular
^^,^ instead of a continuous circiilar area, then
lTl~n representing ita infernal radius by p,, and
MjL^ii observing that ita area is represented by
<|^^^5j>i *(Pi''— PiOi ^''^^ therefore the pressure upon
XX^^^^^S each unit of it by -7—5 jr, and the fric-

tion of each such unit by -7 , ,., we obtain, as before,

•^ *(p. -P>)
for the friction of each elementary anniilua the

of all the elements of tlie pivot ~„ — ^ / -u or

Let r represent the mean radius of the pivot, 4. e. let
r=i{p. + f^ ; and let I represent one half the breadth of the
r'mg,^.e. let Z=-^(pi—^,); therefore f^=r + l and f^=:r~l.
These values of p, and p, being eutatituted in the above for-
mula, it becomes

1 {r+iy-(r-ir i
oi-\T/f|l+i(i)'j (185.);

whence it follows that thefHeti(m of an anrnda/r pivot pro-
duces the same effect as though the whole pressure were eol-

lected over a point in it distant hy r\ l+-|i ^1 r from ita

centre, where r r^esent its mean radvus and I one half iis
' ■"' From tnia it may be shoM'u, as before, that the

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■wLole work expended upon eacli complete revolution of the
annular pivot is represented ty the formula,

2-/''|l+i(;)'|w (186.)

178. To DEIEKMINE TOE MODtrHJS OF A BTSTEM OF TWO PRES-
SURES APPLIED TO A BODY MOVEABLE ABOUT A FIXED AXIS,
WHEN ITIE POINT OF APPLICATION OF ONE OF THESE PEES-
SDEE8 IS MADE TO EEVOLYE WITH THE BODY, THE PERPEN-
DICULAR DISTANCE OF ITS DIBECTIOS FBOM THE CENTEE RE-
MADilKG OOKSTANTLY THE SAME.

Let the pressures P, and P^, instead of retaining; constantly

^p (as we have hitherto supposed tneni to doj

.-^^X^ ' *^^ same relative positions, be now conceived

'^ A contiuuallj to alter their relative petitions by

i ®" "^ the revolution of the point of application of

P, with the body, that pressure nevertheless

retaining constantly the same perpendicular

distance a from the centre of the axis, whilst

^ the direction of Pj and its amount remain

constantly the same.

It is evident that as the point A, thus continually alters its
position, the distance A,A, or L will continually change, so
that the value of P, (equation 158.) will continually change-
Now the work done under this variable pi'essure during one
revolution of Pj is represented (Art. 61.) by the formula

U „ A*,([idfl, if fl represent the angle A,OA described at

0

any time about 0, by the perpendicular CA,, and therefore
oi,S, the space 8 described in tlie same time by the point of

application A, of P, (see Art. 62.).

Substituting, therefore, for P, its value from equation

(158.), we have

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9jr

.•.U,^'U, + ^-^-^^V, .Ids (187.)

Let now Pj be assumed a constant quantity ;

How L=A,A,= |«,= + 2«A COS. 4+<P;

«,«,*{( «; + «, f

(^ + ^)'/ { 1 + 2(J + j)"'cosJ } **?fl=

{l. + 1-Vf j 1+ |^+5?j~'cos.a j <?fl nearly,

neglecting powei-s of I — -j.— | above the first, since in all

eaees its value is less than unity. Integrating this quantity
between the limits 0 and 2* the eecona term disappears, so
that

Sir
—^l*Ld^= |A^ + A\*2* nearly ;

.•.PA.-L/M=P.(2„,)(i.;i,)'=,U.(i.+ l,)',

since S^ffj is the space throu^ which the point of applica-
tion of tlie constant pressure P, is made to move in each re-

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Tolution. Therefore by ecLiiation (187), in the case in wliicli
Pj is constant,

tJ,=U, 1 1+ (^. + ~.)*P ™- » ) (188).

179. If the pressure P, "be supplied by tlie tension of a
rope winding upon a drum whose radius is a, (as in the cap-
stan), then is the etfoet of the rigidity of the rope (Art. 142.)
the same as though P, were increased by it so aa to become

Now, assuming P, to be constant, and observing that
U,-2'^P,«,, we have, by equation (187),

Substituting in this equation the ahove value for P„

Performing the actual maltiphcation of these factors, ob-
serving tliat — is exceedingly small, and omitting the term

involving the product of this quantity and "- ^— , we have

"Whence performing the integi'ation as before, we obtain

U,=:U,(l + ?-) jl-^jL + iA^psin.,, j +2orD.

If this equation be multiplied by n, and if instead of U, and
TJ, representing the work done during one complete, revolu-
tion, they be taken to represent the work done through n
such revolutions, then

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g,=-uJl+-) j i + (l+l)*p Bin. <f I +3?MrD .... (189j,
whidi 18 the modulus.

"180. If the cLuantity {-^-i — -\ be not so small that terms

of the binomial expansion involving powers of that CLuar.-
tity above the firet may be neglected, the value of th«

definite integral /LtZfl may be detennined as followe :—
={a^-\-aM < 1—, ' ySin.^ V <?5. Let ^ =/^ ' y,

(a, + a,)/'(l— Fsin.'ay (^fl

=2(ffi,+«,)/(l-S'6in. ■d)'(;fl*=3(ffi, + a,)E,(-i;), where Ji;,(^)

repreaenta the complete elliptic function of the second order,
wnose modulus ia %.* The value of this function is given
for all values of A in a table winch will be found at the end
of tliia work.
Substituting in equation (187), -'

" See Mncye. Mel. art. Def. I»t. theorem 2.

t An approxiiHRte value of Ei(i) is given wlicn k is small by the formula
E,(i)=^(1-fK-'), where K=?|^. (See Eni:y<:. Met. art. Def. Int. equation
(W), 14.)

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THE CAP6TAH.

:.L>U, I l+-(-+l|p sin. 9 j . . . . (190).

181. The capstan, ae used on shipboard, is represented in
the accompanying figure.
It consists of a sohd timber
CC, pierced tlirough the
greater part of its length by
an aperture AD, which
receives the upper portion
of a solid shaft AB of great
strength, whose lower ex-
ticniity i** piolonged and
stiongly fixed into the tim
! cr film g ot the ship The piece GG into the upper por
tion of which aie iittel the moveable
aims ot the eapstm tu ni upon the shait
AB resting its weight upon the crown nl
the ihaft coiling t! e cable round its cen
tral portion GG and austaamn^ the ten
s on of the cable by the lateial resi<it4nce
of the shift ThuH tlie caj stan combmea
the rcsiatinces ot the jivot and the ash
ei that the whole leeistance to its motion
1 ci the leeistances due se^aiatel^ ti the
axis'and the pivot, and tlie whole woik expended m tuining
it equal to the whole work which would be expended in
turning it upon its pivot ■\Fere there no tension of tlie cable
upon it, added to the whole work necessary to turn it upon
its axis under the tension of the cable were there no friction
of the pivot, Now, if U, represent the work to be done
upon the cable in n complete revolutions, the work which
must be done upon the capstan to yield this work upon iha
cable is represented (eijuation 189.) by

(1+5.) |l-|-(4 + iyp8in.(p.lu,+2wrD,

B equal to the

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■where q, represents the length of the arm, and a, the radius
of that portion of the capstan on which the cahle is winding.
Moreover (Art. 176.), the work due to the friction of the

pivot in n complete revolutions is represented by o'^'^P^/^-

On the wliole, therefore, it appears that the work U,
expended upon n complete revolutions of the <
represented by the formula

U,+

2w*{d + |./w}...(191).

which is the modulus of the capstan.

A single pressure V, applied to a single arm has been
enpposed to give motion to the capstan ; in reality, a num-
ber of such pressures are applied to its different arms when
it is used to raise the anchor of a ship. These pressures,
however, have in all cases,— except in one particular case
about to be described,~a single resultant. It is that single
resultant which is to be considered as represented by P,,
and the distance of its poiut of application from the axis
by a„ when more than one pressure is applied to move the
capstan.

Tlie pai'ticalar case spoken of above, in which the pree-
sm-es applied to move the capstan have no resultant, or can-
not be replaced by any single pressure, is that hi which
they may be divided iato two sets of pressure, each set hav-
ing a resultant, and in which these two resultants are equal,
act in opposite directions, on opposite sides of the centre,
perpendicular to the same straight line passing through the
centre, and at equal distances from it.*

Suppose that they niay be thus compounded into tlie
equal pressures E, and K,, and let them be replaced by
these, lie capstan will then be acted upon by four pres-
sures,— ^the tension T, of the cable, the resistance R of the
shaft or axis, and the pressures "R^ and B,. liJ"ow these pres-
sures are in equUibnum, If moved, therefore, parallel to
their present directions, so as to be applied to a single point,

* Two equal pressures thus placed constitute a sTiriOiL cotrpLS. The pro-
pertiea of such couples have been fuUv discussed by M. Poinsot, and by Mr.

'Pi.iti^tiDp.^ m lti« Ti-p»f.iqA fin E^IHtioAl rfnimli'Q : f^nnu: fli^pniint iif thpin will ha

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196 TnE CAPSTAN.

they would he in equilibrium about that point (Art. 8.).
But wlieu so remoTed, E, and ~R^ ■will act in the sam,e
strmgkt Une and in opposite directions. Moreover, they
are equal to one another ; R, and Ej will therefore sepa-
rately be in equilibrium with one another when applied to that
point; and therefore Pj and R will separate^ha in equili-
brium ; whence it follows, that R is equal to P, or the whole
pressure upon the axis, equal iu this case to the whole tension
P, upon the cable. So that the friction of the axis is repre-
sented in every position of the capstan by P^ tan. ip {tan. 9
being equal to the co-efflcient of friction (Art. 138.)3, and
the wori expended on the friction of the axis, whilst the
capstan revolves through the angle S by P^pfl tan. 9, or by

P,B,S I— j tan. », or by l^i(~) *^^" ''' ' ^** ^^^^' ^^ ^^ whole,

introducing the correction for rlgi&l-by and for the friction of
the pivot, the moduhis (equation 191) becomes in this case

U,=U,(l+^) jl+(^)tan.9} +

2»MrJD+|p,/wl....(192>

This is manifestly the least possible value of the modulus,
being veiy nearly that given (equation 191) by the value
infinity of a,.*

Thus, then, it appears generally from equation (191), tliat
the loss by friction is less as a, is greater, or as P, is applied
at a greater distance from the axis ; but that it is least of all
when the pressures are so distributed round the capstan as
to be reducible to a couple, that ease corresponding to the
value infinity of a,. This case, in which the moving pres-
sures upon the capstan are reducible to a oo'itple, manifestly
occurs when they are aiTanged round it in any number of
pairs, the two pressures of each pair being equal to one an-
other, acting on opposite sides ot the centre, and perpendi-
cular to the same line passing through it. This symmetrical
distribution of the pressures about the axis of the capstan is
therefore the most favourable to the working of it, as well
as to tlic stability of the shaft which sustains the pressure
upon it.

* ^ being exceedingly small, tan. ^ is very ncsrlj equal to sin. ^.

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82. The modulus of a system of there pkessiireb a
to a body moveable about a cylindeicai. axis, two of

THESE PiiESSUitES BEING GIVEN IK DIEECmON AND PARAL-
LEL TO ONE ANOTHF.K, ASD THE DIKECTION OF THE THIRE
CONTINUALLY ItEVOLVING ABOUT THE AXIS AT THE SAME
PERPENDIOULAR DISTANCE FROM IT.

Let P, and P, represent the parallel pressures of the sys-
p-. torn, and P, the revolvina; pressure.

From the centre of the axis O, let fall
// "^ the pei-pendieulars 0A„ CA„ OA, upon
H / H/ i the directions of the pressures, and let

^ ^^-■if'44 ^ represent the inclination of CA, t«

i 'W-^,Y CA, at anyperiod of the revolution of

jp, ai *^ P,. Let P, be the preponderating

pre^ure, and let P, act to turn the
Bjstem in tlie same direction as P„ and Pj in the opposite
direction ; also let E represent the resultant of Pj and P,,
and r Uie perpendicular distance CA of its direction from 0.
Suppose the pi'essures P, and Pj to be replaced by E ; the
conditions of the equilibrium of P, throughout its revolu-
tion, and tlierefore the worh of P, will remain unaltered by
this change, and the system will now be a system of two
pressures P, and II instead of three ; of which pressures It
IS given in direction. The modulus of this system is there-
fore represented (equation 187) by the formula

■U^=Vr+^-^—J'A .Zdi (193) ;

where Ur represents the work of 31, and L represents tlie dis-
tance AA, between tlie feet of the perpendiculars r and a„
so that U':^a'—2a,r <ios.&+r':={aj—rcoB.S)'+r' SULK'S;

;. EX'=(Ila,— Rr cos. fl)' -f-KV sin.'^.

Now, E=P,+P„ E?'=Pa-Pa ;

.■.E=L'=j(P,+P>.-(P,«.-P,a.,)cos.sp-l-(PA-PA)°siii.'a,

[Now if the relations of a^ to a, are such that

j (P,-|-P,)«,-(Pa-Pa) cos. 6 \ >(PA-PAT8in.'fl

then the value of WL' will be represented by the sum of the

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scLuares of two quantities the first of which is greater than
the second. Ed.] Therefore, extracting the square root by
Poncelet's theorem, (see Appendix B.)

RL=«{(P.+P>,-(P,t(,-pA)co8.a}+/3(P^a_P,«,) Bin. a
very nfearly ; or,
KL=aa,(I'= + P.)-(FA-PA)(°coe.S-,Ssin.fl). . . .(194).

/(P,ffi,-P,»,)(« COS. S-/3 sin. ^)c

J'KLdfi=m, I ^" + ^' I -y"(PA-PA)

(« cos.l-^ ein. S)S. . . . (195).
If P, and i*, be eonstami, the integral in the second raenihei'
of this equation becomes (P.ffl,— P,».) (a sin. 3 + ^ cos. fl);

whence observing that P,a,—P,t(,~— -■■■;■ ■ =— ^t — ^;

also, that JJ^=S'S.r^6^,a,—Sp^a^~V,~V„ and substituting
in equation (193), we have

TJ,=TT,-U,+p eui. <p j a (-' + -') -

jEi^'Wasiu. fl + /3co3.fl)l. . . . (196);
for complete revolution making fl=2*, we have

u.='n

reducing,

v.=v.^v.^,.^.,Ul-^^]-,0^]

which is the modtdus of the system where a and 5 are to be
il','! cnuined, as in Note B, (Appendix.)

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THE OHINKSK CAPSTAN. 199

18S. If the pressure P, be supplied by the tension of a cord
which winds upon a cyliuder or drum at the point A„ then
allowance must he made for the rigidity of the cord, and a
correction introduced into the preceding equation for that
purpose. To make this correction let it be observed
(Art. 142.) that the effect of the rigidity of the cord at A, is
the same as though it increased the tension there from

or (multiplying both sides of this inequality by a„ and inte-
grating in rcapeet to fl,) as though it increased

El

or, U, to(l+~tTI, + 2*D.

Thus the effect of the rigidity of the rope to which P, is ap-
plied upon the viork TJ, of that force is to increase it to

|H — jir, + 2ffD. Substituting this value for TX, in equa-
tion (197), and neglecting terms which involve products of

the exceedingly small quantities—,^ '■ — ,- '- — ,andD,

we have

] 1-p sm.»(l+ A) I TJ, + 2.D. . . . (198).

To determine the modulus for n revolutions we must sub-
Btitute in this expression nt for *.

The Chinesb Cafstan.
18i. This capstan is represented in tlie accompanying

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THE CHIMESE CAPSTAN.

figure under an exceed-
ingly portable and con-
ij venient form.* The axle
or drum of the capstan
ia composed of two parts
of different diameters.
One exti-eniity of the
cord is coiled upon one
of these, and the other, in an oposite direction, upon the
other ; so that when the axle ia turned, and the cord is
wound -upan one of these two parts of the drum, it ia, at the
same time, wound oj' the other, and the intervening cord ia
shortened or lengthened, at each revolution, by as much ae
the circumference of the one cylindta" exceeds that of the
other. In thus passing from one pSrt of the drum to the
other, the cord is made to pass round a moveable pulley
which sustains the pressure to he overcome.

To determine the modulus of this machine, let u, and «,
represent the work done upon the two parts of the cord
respectively, whilst the work U, is done at the moving point
of the machine, and TJ, yielded at its working point.

Then, since in this case we have a body moveable about a
cylindrical axis, and acted upon by three pressures, two of
which are parallel and constant, viz. the tensions of the two
parts of the cord ; and the point of application of the third
IS made to revolve about the axis, remaining always at the
same perpendicular distance from it ; it follows (by equation
198), that, for n revolutions of the axis,

U,=Aw,— B«,+2»M-D (199) ;

where

A=h + - + psin.<p{-_-£— \ Land

«5 and a, representing the radii of the two parts of the dram,
tf; the constant distance at which the power is applied, and p
the radius of the axis.

• A figure of the capstan with a double aile was seen by Dr. 0. Gregoi-y
among some Cliinese drawings more tliau a century old. It appears to nava
been iuTentjid under tlie particular form shown in the above figore by Mr. G.
Eokhardtand by Mr. M'Lean of PliiladelphJa. (See Professor Robinson's JfteA.
I'hiL vol. ii. p. 255.)

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THE CHISE5E CAPSTAN.

Alao, Biaeo- the two parts of the cord pass over a pulley,
and the pulley is made
to revolve under the ten-
sions of the two parts of
tile cord, m, temg the
■work of that tension
wliich preponderates, we
haTe (by equation 181),
if S represents the lengtli
of cord which passes
over the pulley,

B,

l)t

1+^+

2 Wcos.i\

D

Pi 6in. 9 \

a representing the radius of the pulley, p, tlie radius of its
axis, W its weight, and j the inclination of the direction of
the tensions of the two parts of the cord to the vertical, the
axis of the pulley heing supposed horizontal, and the two

■parts of the cord parallel, Now t,= — ^, t= — 5~. Snb-

stituting these values, and multiplying by 2nir»„ we have

— ?-^:=A,Mj-|-2wrtfjB, .

. (200).

Since the tensions t, and f, of the two parts of tlie cord,
and the pressure P, overcome by the machine, are pressures
applied to ihe p'ulley and in equilibrium, and ihat the points
of application of t, and P, are made to move in directions
opposite to those in which those pressures act, whilst the
pomt of application of *, is made to move in the same direc-
tion ; therefore (Art. 59.),

Eliminating u^ and ii^ between this equation and equation
(200), we have

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THE nOESE

Substituting these values in eq^uation (199). and reducing,

a. I

Substituting their values for A, A„ B, B„ neglecting t«rm9
involving more than one dimension of — '—, — , &c. and

reducing, we obtain for the

of the machine,

a.1 'i.naj!

|E^+«(l-r-jpsin.?i|D+Wp,co8.isin.(p}
a(l-^j+E + 3p,ein.(p

2n*..(201).

From which expression it is apparent that when the radii ct,
and », of the double axle are nearly equal, a great sacriiice
of power is made, in the use of this machine, by reason of
the rigidity of the cord.

The Hokse Capstak, oe thk 'Wnm Gm.

185. The ■whim is a form of the capstan, uaed in tlie jwst
op&raU<yiis of mining, for raising materials from the shaft and
levels by the power of horses, h^ore the quantity excavated
is so great as to require the application of steam power, or
before the valuahle produce of the mine is sufficient to give
a retui-n upon the expenditure of capital necessaiy to the
erection of a steam engine. Tlie conetmction of this machine
will he sufficiently understood from the accompanying figure.
It will ta ohserved that there are two ropes wound upon tlie
drum in opposite directions, and which ti-averse the space

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THE H0K8E GAP6TAN. 203

between the capstan
mil the movith of the
shatt One of these
c allies at its extrem-
ity the descending
, (emj t^ ) bucket, and is
cojit nuahy in the aet
of ■winding, off the hum >f the capstiii t^ it revolve ; whilst
the othei liom whose oxtiemit; is suipended the ascending
goaded) bncket, contmually -winds on tlie duim. The pres-
snre exerted by the horses is that necessary to overcome the
friction of the different bearings, and the other prejudicial
resistances, and to balance the difference between the weight
of the ascending load, bncket, and rope, and that of the
descending bncket and rope. The rope, in passing from tlie
capstan to the shaft, traverses (sometimes for a considerable
distance) a series of sheaves or pulleys, such as those shown
in the accompanying ligm-e.

Let now a, represent the radius of the drum on which the
rope is made to wind, and n the nnmber of revolutions
which it must make to wind np the whole cord ; also let f*
represent the weight of each foot of cord, and & the angle
which the capstan nas described between the time when the
ascending bucket has attained any given position in the
shaft and that when it left the bottom ; then does a^i repre-
sent the length of the ascending rope wound on the drum,
and therefore of the descending rope wound off it. Also,
let W represent tlie whole weight of the rope ; then doss
"W— fia,^ represent the weight of the aacmmna rope, an<l
V-a,& that of the descending rope, both of which hang sus-
pended in the shaft. Let P, represent the load raised at
each lift in the bucket, and w the weight of the bucket ;
then is the tension upon the ascending rope at the mouth of
the shaft represented by "W"— (i»,^+Ps+w, and that upon
the descending rope by iJ-a,6+w.

Let, moreover, p, and p, represent the tensions upon tliese
ropes after they have passed from the mouth of the shaft,
over the intervening pulleys, to the cheumference of the
capstan.

Now, since the tension upon the ascending rope, which is
'W—f-ctji + J^^+to at the mouth of the abaft, is increased to
_^s at the capstan, and that the tension upon the descending
rope, which is jfj at tlie capstan, is increased to y-a^i+io at
the mouth of the shaft, if we represent by (1 -|- A) and B the
constants which enter into equation 180 (Art. 174.), we have,

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THE HOEBB CAPBTAH.

by that equation (oheerving that U^^PiS, and C,=:PjS,. so
that S, disappears from both sides of it),

p,={l + A){W+V,+w-!^a,S)+B, .... (203),

and (iffi,fl+«'=(H-A)^,+B .... (203).

Multiplying the former of the ahove equatioBs by 1 + A,
adding them, transposing, dividing by (1-1- A), and neglec^
ing terms of more than one dimension in A and B,

^,-^,=(l+A)(W+P,}+3AMH-2B-2na,l

INow IT,, in eqaation (193) represents the work of the
resultant of p, ana ^, dniing « revolntions of the capstan, it
ttierefore equals the difference between the work of p^ and
that of _^, (eee p. 198),

:.\Sr=Jp,a,di -JpAdi = aJlp,-p,)dd;

.•.V-=a,^\{-l+A){^V + 'P,) + 2Aw + 2B-'2!^a,e\de~

{(H-A)(W+P,)+2Aw+2BK3»w»,)-fA(2W(t,)';
.■.lj^=(l + A)ir,+ Kl + A)'W"+2Aw + 2B-[iSJS,..(204:) ;

observing that 2mTa,=S„ and that P,Sj=XTj.

I^^ow, let it be observed that the pressures applied to the
capstan are three in number ; two of them,^;, and p^, being
parallel and actmg at equal distances a, from its axis ; ana
the third, P„ bei^ made to revolve at the constant distance
a, from the asis (Pj representing the pressure of tlie horses,
or the restdtant of the presswee of tne horses, if there be
more than one, and a, the distance at which it is applied) ;
60 that equation 193 (Art. 183.) oitams in respect to the
presBurefl P„ p^, p, ; tt, beingassumed equal to a,.

Substituting _Pj and^^ for P, and P, in equation (19i),

'BL=a.a,{p,+p^—a,{p^—p^) (n. cos. 0—0 sin. fl) ;

.-. fjiLde=^a,f{p,+p,) d9-a,f{p~p,)
(a COS. 0—0 sin. ff) dfi.

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THE HOESE CAPSTAN.

Now, tile termg of eq^uation (180), represented in tlie above
equations by A and B, are all of one diinenaioii in the exceed-
ingly email qnantities D, E, sin. 9. If, thei-efore, the valHes
of _p, and _^3 given by those equations be substituted in the

value of P s'^i- '^.fliUlo (equation 193), then all the terms

of that expression which involve the quantities A and B will
be at least of two dimensions in D, E, siu. 9, and may be ne-
glected. Neglecting, therefore, the values of A and B in
equations (202, 203), we obtain

^^+j),='W-|-P,+2w, andi)3-^,=W+P,— 2(ia,e ;

.-. aj'{j},+p,)de=a, |W+P, + 2^2n*=(lA{(2?i^a,)P,

+ i2mfa,){W + 2w)\= (^\ {S,P,+S,CW+2w)i

representing by S, the
space described by til e
load, and by XT, the
usefulworli done upon
it, daring n revolu-
^^ tions of tlie capstan.

Similarly,

SnT SnTT

a,f{p,~p:)iciCos. e- 13 Bin. ef)d0 = a, f{'W+V,~2!>.a,0l

(a COS. e-!3 sin. e)de=a.lW+T,) /*(« cos. 6-/3 sin. ff)de~

Sca.'A" COS. (9-,S sin. 6)dd9.

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THE HOKSE CAPSTAK.

Xow/ {<•■ COS. e—13 sin. 6)dd=^, and Ha cos. 0~li sin. 6)

6d6=2finv^;

••■ (^■.fip.-p^i?- COS. fl-/3 sin. e)dd=^a,{^+V,)-1^i>.a,

(27Ka,)=,Sa,^+^«,(W-%S,); observing that P,=5>

.■.y*EL<?e=a(^)|U,+S,(W+2y.)}-/3a^^_/3«,{W-2f.S,);

Substituting tliie value, and also tliat of TJ,. (equation 204)
in equation (193), and assuming

C, = (l+A)W+2Am + 2B and 0,= t^ + iw)lii\ +2ri„

we have

U,=(l + A)U,+C,S,-nS,'+fciM

9 8 e

' For /*9coE.M0=9sin.0-/'sia.fl(rP*=0Hm.fl-Tera.e; also A sin. fl*?
0 0 c

= — 0 COS. fl + /ooa.M9*=-0EOS.fl+sin.e. Xow, substituting annfore,

0
tieBe integrals become respecUvely 0 and —Imr.

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\ ' a, ) '' a^ '

■which is the modulus of tlie machine, all the varlone ele-
ments, whence a sacrifice of power may anse in the working
of it, being taken into account.

The Ekiction of Coeds.

186. Let the polygonal line ABC . . . YZ, of an infinite

^ , ^^t;. number of sides, be taken to represent

"^^^^^^ K ^*^~^ the ctuved portion of a cord emtiraciiig
■""■•-.^ \ \j^;.i ^'^ any ai'c of a cyluidrical surface (whe-
T %X''\ 'ifc ther circulftT or not), in a plane per-

^'^'' >,i "■ pendicular to the axis of tlie cylinder ;

*v i\ also let Aa, B5, Co, &c., be normals

\^p V or perpendiculars to the curve, inclined

' * to one another at equal angles, each

represented by ^9. Imagine the surface of the blinder to
bo removed between each two of the points A, B, &c., in
succession, so tliat the cord may be supported by a small
portion only of the surface remaining at each of tlioae
points, whilst in the intermediate space it assumes the direc-
tion of a straight lino joining them, and does not touch the
surface of the cylinder. Let P, represent the tension upon
the cord before it has passed over the point A ; T, the ten-
sion upon it after it has passed over that point, or before it
passes over the point B ; Tj the tension upon it after it has
passed over the point B, or before it passes over 0 ; T, that
after it has passed over C ; and let P, represent tlie tension
upon tlie cord after it has passed over the nth or last
point Z.

Now, any point B of the cord is held at rest by the ten-
sions T, and T, upon it at that point, in the directions EG
and BA, and by the resistance R of tlie sinface of the cylin-
der there ; and, if we conceive tlie cord to be there in the
state bordering upon motion, then (Art. 138.) the direction
of this resistance R is inclined to the perpendicular 6B to
the surface of the cylinder at an angle KB& equal to tlie
limiting angle of resistance ip.

./Google

Now T„ Tj, and E are pressures in equilibrium : tliere-
fore {Art, li.)

T,_sin. T^R
T,~sin. T.BR'

biitT,BE=AE&-EB5=i(*-AfflB)-EB5 = 5-^-?,

■\1~{j-^)\_'''-{y-'^)

T, . (< /A6 \ ) /AS \

. T,~T, .

or dividing numerator and denominator of tlie fraction in the

AS

second member by cos. -^ cos. p,

Suppoae now the angles Aab, B&C, &c., eacli of which
ualB^' ■ ' ■" ■■ ■■" --'■■■'--■'--- ■-1 -----

eqwafs Afl, to be exceedingly small, and tlierefore the points
A, B, C, &c., to be exeeedinely near to one another, and
exceedingly mimeroue. By tiiis supposition we shall mani-
festly approach exceedingly near to ttie actual case of an inr
fin/ite number of such points and a continuous surface ; and

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TIIE FlilCTION OF COEDS, 209

if we suppose M infinitely small, our supposition will coincide
with that case, !Now, on the supposition that ^iS is exceed-

Ad
ingly small, tan. -5- . tan. <p is exceedingly small, and may

i compared with unity ; it may thei'efore he
i in the denominator of the above fraction. More-

over A3 being exceedingly small, tan. -x- = -^

T-T
.-. ■ 'y - ■ = tan. (p . Afl* ; .-, T,=i; (1+ tan. <? . Afl).

Now the number of the points A, B, 0, &c. being repre-
sented by n, and the whole angle AdZ between tlie extreme
normals at A and Z by S, it follows (EucHd, i. 33,) that

6:r=n. Afl; therefore Ad=- ;

Similarly, P>=T. {i+-tan. 9)

T,=T. (l+-tan. 9),

T^i=Pai+-tan.<p).

Multiplying these equations together, and striking out fac-
tors common to both sides of their product, we have

P,=P,a + ^an. ?)";

■ If we coiiaidec the tension T as a function of 0, of which any ci
values are represented by Ti and Ta, and their differeneo or
Tby AT, thon^^^=;tRn. 0. Afl; therefore S ■ "To — —tan. *; therefore,
pasang to the limit s jo = — *a"- ^\ "■>'* iitUgroimg between, the lindts 0
and ft obsemng that at the ktter limit T=Pi, and that at the former it equala
Pi, we hare log. j — )=- Stan. 0; therefore Pi=Pie ""'■*.
14

./Google

THE FRICTION OF COllDS.

orP,=P, 1+a tan. 9 + -^S" tan. '<p +

-^ 5— Ctan. 9+...}.

Now this relation of P, and P, obtains however small A*
be taken, or however greai n be taken. Let n be taken
iwfimtdy great, so that the points A, B, 0, &c. may be
infinitely numeroua and infinitely near to each other. The
gv^osed case thus passes into the actual case of a con-

tinnouB surface, the fractions -

above equation becomes

But the quantity within the brackets is the well kiiown ex-
pansion (by the exponential theorem) of tlie function eS'^n-it,

.-. P^-P^ee'^-'S* (205).

Since the length of cord S„ which passes over the point
A, is the same with that 8, which passes over the point Z,
it follows that tlio modulus (Art. 152.) of such a cylindrical
surface considered as a machine, and supposed to he fixed
and to have a rope pulled and made to shp over it, is

U.=TI,Efl*'"'-0 (206).

It is remarkable that these expressions are whoUy inde-
pendent of the form and dimensions of the surface sustain-
mg the tension of the rope, and that they depend exclu-
sively upon tlie inclination i or AdZ of the normals to the
points A and Z, where the cord leaves the surface, and upon
the co-ef&cient of fi-ietion (tan. ?), of the material of which
the rope is composed and the material of which the surface
is composed. It matters not, for instance, so i'ar as the_/Wo-

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THE FKIOTIOS OF COKDB. 311

tion of the rope or band is concerned, wliether it passes
over a large pulley or drum, or a small one, provided the
angle subtended by the arc which it embraces is the same,
and the mateaiala of the pulley and rope the same.

In the case in which a cord is made to pass m times round
snch a surface, fl=2m7r ;

And this is true whatever be the form of the surface, so
that the pre^ure necessary to cause a cord to slip when
wound completeh/ round such a cylindrical sm-face a given
number of times is the same (and is always represented by
this quantity), whatever may be the form or dimension of
the surface, provided that its material be the same. It
matters not whether it be squai'e, or circular, or elliptical.

187. If P,', P/', P/", &c. represent the pressm-es which
must be applied to one extremity of a rope to cause it to
slip when wound once, twice, three times, &c. round any
such surface, the sams tension P, being in each case sup-
posed to be applied to the other extremity of it, we have

P/=P,£^""''ft, P/'=P,«*'rH"-0, P,"'=P,e«Jr'«>''^, &c.=&c.

So that the pressures P,', P,", P,'", &c. are in a geome-
trical progi"ession, whose common ratio is e^irtan.^^ which
ratio is always greater than unity. Thus it appears by the
experiments of M. Morin (p. 135.), that the co-efHcient of
friction between hempen rope and oak free from unguent is
■33, when the rope is wetted. In this case tan. ip=-33 and
Sn-tan. (p=2x 3-14159 x-S3=3-0T345. The common ratio
of the progression is therefore in this case e^'Ois^^ or it is the
number whose hyperbolic logarithm is 2'0'i'345, This num-
ber is 7'95 ; eo that each additional coil increases the fric-
tion nearly eight times. Had the rope been dry, this
proportion would have been much greater. If an additional
Afl^eoil had been supposed continually to be put upon the
rope instead of a whole coil, the friction would have been
found in the same way to increase in geometrical progres-
sion, but the common ratio would in this case have
been eTtan.^ instead of e^ir"™-^. In the above example the
value of this ratio would for each kcdf coil have been
2-83.

The enormous increase of friction which results from

./Google

THE FRICTION (

eacli additional turn of tho cord upon a capstan ov dvnm,
maj from these results be understood.

188. We maj, fi'ora what has been stated above, readily
explain the reason wliy a knot connecting the two extrerai-
ties of a cord effectually resists the action of any force
tending to separate tliem. If a wetted cord he wound round
Fifj. 1. m. a. Fig. 3. a cylinder of oak as

in Jig. 1., and ita ex-
tremities he acted
upon by two forces P
and E, it has been
shown that P will not
overcome K, unless it
be equal to some-
where about eight times that force. Now if the string to
which R is attached be brought nnderaeatii the other stiing
BO as to hu pressed by it against the surface of the cylinder,
as at vn^Jhg. 2.; then, provided the friction produced by
this pressure he not less than one ei^th of P, the string will
not move even although the force li cease to act. And if
both extremities of the string be thus made to pass between
tlie coil and the cylinder, as in fig. 3., a still less pressure
upon each will be requisite. Now, by diminishing tlie
radius of the cylinder, this pressure can he increased to any
extent, since, by a known property of funicular curves, it
varies inversely as the radius.* We may, therefore, eo far
diminish the radius of a cylinder, as that no force, however
great, shall be able to pull away a rope coiled upon it, as
represented in fig. 3., even although one extremity were
loose, and acted upon by no force.

^'i'-*- Let us suppose the rope to be

doubled as in fig. 4., and coiled
as before. Tlien it is apparent,
from wliat has been said, that
the cylindei miy be made so
sraill, that no lorces P and P'
applied to the extremities of
either ot the double cords will
bo "^ufiicipnt to pull them from
it, in whatever directions these are applied

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THE FRICTION BKEAK. 213

Now let the cylinder be removed. The cord then lieiTig
drawn tight, Lisfdad of being coiled round the cylinder, will
be coiled round portions ot itself, at the points m and n ;
and instead of being pressed at those pointe npon the cylin-
der, by a force acting on one portion of its circumference, it
■will be pressed by a greater force acting all round its cir-
cumference. All that has been proved before, with regard
to the impossibility of pulling either of the cords away from
the coil when the cylinder is inserted, will therefore now
obtain in a greater degree ; whence it follows that no forces
P and P' acting to pull the extremities of the cords asunder,
may be suiBcient to separate the knot.

The Prictioh Bkeak.

189. Tliere are certain machines whose motion tends, at
certain stages, to a destnictive acceleration ; as, for instance,
a crane, which, having raised a heavy weight in one position
of its beam, allows it to descend by the action of gravity in
another ; or a railway train, which, on a certain portion of
its hne of trai^t, descends a gradient, having an melination
greater than the limiting angle of resistance. In each of
fliese cases, the wort done by gravity on the descending
weight exceeds the work expended on the ordinary resist-
ance due to the friction of the machine ; and if some other
resistance were not, under these circumstances, opposed to
its motion, this excess (of the work done by gravity upon it
over that expended upon the friction of its rubbing surfaces)
would be accumulated in it (Art. 130.) under the form of
vis viva, and be accompanied by a rapid acceleration and a
destructive velocity of its moving parts. The extraordinary
resistance required to take up its excess of work, and to
prevent this accumulation, is sometimes supplied in the
crane by the work of the laborer, who, to let the weight
down gradually, exerts upon the revolving crank a pressure
in a direction opposite to that which he used in raising it.
It is more commonly supplied in the crane, and always in
the railway train, without any work at all of the laborer, by
a emva^e pres&v/fe of his hand or foot on the lever of the fric-
tion bre^, which useful instrument is represented in the
accompanying figure under the form in which it is com

./Google

2ti THE FRICTION BREAK.

monly applied to the eiane, — a foim of it which may serve
to illustrate the pimLiple ot its applicai n uiider every
othei BC represents a wheel
fi\ed commonly upon that
axib ot the machine to ■which
' tlio crank ib attached, and
wliieh axis is earned round
by it with gieitei velocity
than any other The pen-
phen ot thi& wiieel, which is
usually of cast iron, is em-
braced by a strong band* ABOE of wrought iron, fixed
fiiinly by its extremity A to the frame of the machine, and
by its extremity E to me short ai-m AE of a bent lever PAE,
■which turns upon a fixed axis or fulcrum, at A, and whose
arm PA, being prolonged, carries a counterpoise D just
sufficient to overbalance the weight of the ai'ni AP, and to
relieve the point E of all tension, and loosen the strap from
the periphery of the wheel, when no force P is applied to the
extremity of the arm AP, or when the break is out of
action.

It is evident that a pressure P applied to the extremity of
die lever will produce a pressure upon the point E, and a
tension upon the band in the direction ABOE, and that
being fixed at its extremity A, the band will thus be tight-
ened upon the wheel, producing by its frietion a certain
resistance upon the circumference of the wheel.

Moreover, it is evident that this resistance of friction upon
the circumference of the wheel is precisely equal to the
tension upon the extremity A of the baud, being, indeed,
wholly home by that tension ; and that it is the same
■whether the wheel move, as in this case it does, under the
band at rest, or whether the band move (under the same
tensions upon its extremities, but in the opposite direction)
over the wheel at rtst. Let R and Q represent the tensions
upon the extremities A and E of tJie band ; then if we sup-
pose the wheel to be at rest, and the baud to be drawn ovei'
it in the direction EOB hy the tension E, and i to represent
the angle subtended at the centre of the wheel by that part
of its circumference which the band embraces, we have
(equation 205)

./Google

Let a, represent the length of tlie ai-ni AP, and a^ the
length of the perpendiculai- let fall from A upon the direc-
tion of a tangent to that point in the circumference of tJie
wheel where Uie end EC of the hand leaves it.

Then, neglecting the friction of the axis A, we have
(Art. 5.)

. (20T).

If Pj represent any pressure applied to the circumference of
the break wheel, and Pj a pressure applied to the working

Eoint of the machine, whatever it may be, to which the
reak is applied, and if P,=aP,+J (Art. 152.) represent the
relation between Pj and P, in the i/nferior state bordering
upon motion by the preponderance of P, ; then, when P, is
taken in this expression to I'epresent the pressure W, whose
action upon the working point of the machine the break is
intended to control, P, will represent that value R of the
friction upon the break whicli must be produced by the
intervention of the lever to control the action of the pressure
W" upon the machine ; so that taking E to represent the
same quantity as in equation (207), we have

E=aW + 5.

Eliminating E. between this equation and equation (207).
and solving in respect to P,

^(a.W+%-

The Band.

1 *0. "When the circular motion of any shaft in a machine,
and the pressure which accompanies that motion, conati-
tuting together with it the icorh of the shaft, are to be com-
municated to any other distant shaft, this commnnication is

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210 THE BAND.

usually established by means of a band of
leather, which passes round drums fixed upon
the two shafts, and has its extremities drawn
together with a certain pressure and united,
eo as to produce a tension, which should be
just that necessary to pre%'ent the band from
slipping upon the drums, subject to the pres-
sure under which the work is transfen'ed.
The faeiUty with which this communication
of rotatory motion may he established or
broken at any distance and under almost
every variety of cu'cumstance, has brought
the band so extensively into use in machinery,
that it may be considered as a principal chan-
nel through which work is made to flow in its distributiou
to the successive stages of every process of mechanism,
carried on in the same workshop or manufactory.

191, The smn of iht. tendons wpon the two jparU of a hwnd
is the same, whatever he the^essv/re imder which the Txmd
is drwen, or the resistcmce overcome, the tension of the
drivingpart of the hahd b&ing always increased hyjust so
mMoh as that of the dfivenpart is dimamshed.

This principle was first given by M. Poncelet ; it has since-
been amply confirmed by the experiments of M. Moiin.* It
may be proved as followsf : — In the very eommencement of
the motion of that drum to which the driving pressure is
applied, no motion is communicated by it to the other drum.
Before any such motion can be communicated to the latter,
a diff&r&nce must be produced between the tensions of the
two parts of the band sufficient to overcome the resistance,
whatever it may be, which is opposed to the revolution of
the driven dnim. Now, an increase of the tension on the
driving side of the band must be followed by an elongation
of that side of the band (since the band is elasticV and by
the revolution of the circumference of the drimng drum
through a space precisely equal to this elorigation. Sup-
posing, then, the other, or driven side of the band, to
remain extended, as before, in a straight line between its two
points of contact with the drums, this portiim of the bitud

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THE BAND,

317

must evidently have c(}ntraated by precisely tlie length
through which the circumference of the 3rivmg drum has
revolved, or the driving side of the band elongated. Thus,
the elongation of the diiving side of the band is precisely
equal to the contraction of the driven side. M^ow, the band
being supposed perfectly elastic, the increase or dimi-
nution of its tension is exactly proportional to the increase
or diminution of its length. The ijicrease of tension on the
one side, produced by a given elongation, is therefore pre-
cisely equal to the diminution of tension produced by a con-
t]-action equal to that elongation oti the other side. Thm,
if T represent the tension apon each side of the band before
the driving pressure, whatever it may be, was applied,
and if T, and Tj represent tlie tensions upon the driving
and the driven sides of the band after that pressure is
applied ; then, since T,— T represents the increase of tension
on the one side, and T— T, the diminution of tension on the
other, T."T=T"-T, ;

.•.T,-|-T,=2T ,

It is a gi-eat principle of the economy of power in the use
of tlie band to adjust this initial tension T, so that it may
just be snfBcient to prevent the band from slipping upon
the drum under any preeam-e which it is required to transmit..
'Wie means of making fhia adjustment will '
hereafter.

The Modulus of the Baud,

192. For simplifying the consideration of this important
element in macliineiy, we shall fii'st consider a particular
case of its application. Let the two djrwms,, whose axis are
Ci and C5, be supposed equal to one another, so that the two
parts of the band which pass round them may be parallel.
Mg\. Fig. 2. L,et, moreover, the centres of the

two drums be in the same verti-
cal straight line, so that the two
, parts of the band may be verti-

Let P; and P^ be pressures ap-
plied, in vertical directions, to
turn the dmms, and at perpen-
dicular distances from their cen-
tres, represented by 0,Pi and
C,P, ; of which pressures P, is
the working or driven pressm-e,

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218 THE BAND.

or that ■which is upon the point of yielding by the prepon
deraiice of the other P,. In jig. 1. P, is seen applied on
the same side of the centre ot tne dmma as Pj, and in j?^.
3. on the opposite side. Let T, and T, represent tlie tensions
npon the two parte of the baud, T, being that on the Srimng,
and Tj that on the drwen side.

»,=C,P„ «,=0,P„

r=radiuB of each drum,

■W=weight of each drum,

p=:radins of axis of each drum,

El and It,=re8istanees of axes of drums,

(j>=Umiting angle of resistance.

Now, the parallel pressures P„ "W, T„ T„ E„ applied to the
lower drum, are in eqidUbrmm ; therefore (Art. 16.),

I1, = ±(T.+T,-P.-W);

or substituting for T,+T, its value 2T (ecLuation 209),

E,-:±(3T-P,-W) (310).

The sign ± beuig taken according as 2T is greater or less
than P,+W, or according as the axis of the lower drum
preeses upon the upper sui-face of its bearings, as shown in
Jig. 1., or upon thelower surface, as shown in^^. 2, In like
manner, the pressures P^, "W, T„ t, E,, applied to the upper
dmm, being in equilibrium,

e,=t.+t,tp,+w,

or (equation 209) E,=2TtP.+"W" .... (211),

where the sign ^ is to be taken according as Pj ts applied
on the same side of the axis as P„ or on the opposite side.

Since, moreover, E, and Ej act, in the state bordering
upon motion, at perpendieulai- distances from the centre of
the axis, which are each represented by p sin, <p (Art. 153.),
we have, by the principle of the equality of momenta,

PA+T/=T,^+E,p3in.?)

pA+T,r+E,psin.v=T,rf *-31^),

observing that the resultant of all the pressures applied to
each drum (excepting only the resistance of its axis) must be
such as would alone communicate motion to it in the direc-
tion in which it actually moves, and therefore that the re
sistance of the axis, which is opposite to this resultant, must
tend to communicate motion to the drum in a direction 02}po-
site to tliat in wiiich it actually nioves.

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219

Subtracting the above eq^uatioiis, and transposing,
PA~PA=(K,+It,) p sin. (?.
Substituting tlie values of E, and E, from equations (210)
and (311), we obtain, in the case in whieh the negative sign
of Ki is to be talcen, or in which 2T is less than P,+"W, the
asis C| resting upon the lower siirface of its collar as shown
in fig. 2.,

Pa-Pa=(P,TP,+2W) p sin. ■? ;
and in the case in which the positive sign of E, is to be
taken, 2T being greater than P,+"W, and tho axis 0, press-
ing against the upper surface of its collai-, as shown in^. 1.,

PA-P=a»=(iT"P,TP,)p sin. 9.
Transposing and reducing, we obtain for the relation be-
tween the driving and driven pressures ' '^
respectively,

e two cases

I',=PJ

\a,+?i

a,— pBin. ip
>Tp6in.y

i^.+psi

■ (213),
. (2U),

and therefore (by equation 121), for the moduli in the two

i+(i).i

, 2S,Wp sin, c

4S,Tp am. »
a,+p sin. 6

■ (215),

In all which equations the sign ^ is to be taken according
as Pj is applied on the same side of the line OjOj, joining the
axis as P„ or on the opposite side.

19S. To deterimne the initial tmaionT! upon the iand, so that
it may not sUp ijpon, the sti^afie of the d/ru/m when sub-
Jeeted to the given resistance opposed to its motion hy the
work.

./Google

THE BAKD.

vC^

Suppose tlio maximum resistance which may, during the-
action of the machine, be opposed to tlie motion
, of the drum to be represented by a pressure P
applied at a given distance a from its centre Cj.
Suppose, moreover, that the band has received
such an initial tension T aa shall just cause it to
be on tlie point of slipping when the motion of
the drum is subjected to this maximum resist-
ance ; and let t^ and *, be the tensions upon
the two pai-ts of the band when it is mus
just in the act of slipping and of overcoming the resistance
P. Now, tlie two parts of the band being parallel, it em
braces one half of the circumference of each drnra ; the relar
tion between ?, and t^ is therefore expressed (equation 205)
by the equation

\J

2T (equation 209),

Also, the relation between the resistance P, opposed to the
motion of the upper drum, and the tensions (, and t^ upon
the two parts of the band, wlien tliis resistance is on the
point of being overcome, is expressed (equation 312) by the
equation

Pa + !;,r+E,p sin. (p=^,r;
or substituting the value of E, (equation 211), and ti-anspos-

Pffl+(2Tq::P+W)p sin. <p=((,-*> ;

whence, suhstituting the value of /,— 4, determined aboi
and transposing, we have

P((3!^p sin. 9)+"W[i sin. 9=:2T

I TK. . l"— fan-!

V +'v

, Google

THE I

■_T^i^ [?('^Tf sin. ./.H"VVpsm.^l ^.^IT).

i®>

194. T^3 modulus of the l>and under Us luoni general form..

The accompanjing figure represents aii elastic band pass-
ing rouna drams of unequal radii, the
line joining whose centres 0; and 0,
J is inclined at any angle to the vertical,
and which are acted upon by any
given pressures P, and r„ P, being
Biipposed to be upon the point of giv-
ing motion to the system.

Let T, and T, represent the tensions
upon the two parts of the band, T, be-
ing that on the driving side.
I, o, perpendiculars upon the directions of P, and Pj re-

6„ 6, the inclinations of the directions of P, and Pj to tlie
line 0,0,.

r„ r, the radii of the di'ums.

W,, W, the weights of the drums.

1 the inclination of the line 0,0, to the vertical, andSajthe
inclination of the two parts of the band to one another.

f, p, the radii of the axes of the drams.

fp the limiting angle of resistance between the axis of the
drum and its collar.

E,, R, the resistances of the collars in which (he axes of
the drums turn in the state bordeiing upon motion, or the
resultants of the pressures upon these axes. The perpendi-
cular distances at which these resistances act from the cen-
tres of the axes are (Art. 153.) Pj sin. <p and p, sin. rp. Since
the ;OTessures acting upon the lower drum are T^ T,, P„ W,,
and R„ and that these preesures ai-e in equilibrium, "W", act-
ing through the centre of the axis, and T, and E, acting to
turn the dram in one direction about the axis, and P, and T,
to turn it in the opposite direction ; we have, by the princi-
ple of the equality of moments (Art, 153,),
P,((,+T,A=T/,+E.p, sin. <p.

And since T„ Tj, P,, W,, li, are similarly in equilibrium

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on the upper dinim, W, acting through the cei^tre, and P„
Rj, T, acting to tuvn it in one direction, whilst T, acts to
tarn it in tlie opposite direction,

.■.PA+T,r,+R,p, sin. (p=:T,j',;
.•.P#,-(T,-T>,=R,p, sin. ? )
Pa-(T.-T>,= -R,p, Bin. 9 f ■
letT.-T,=2!;, andT.+T,=3T,

:.V,a,-2tr,='R,p, sin. ■? 1 ,„,.

T,a,-2tr,---R,f, sin. i. f ^''^^>

To determine the valnes of Tl, and R, let the pressures
applied to each drmn be resolved (Art. 11.) in directions
parallel and perpendicular to the line OiC, ; those applied to
the lower dram which, being thus resolved, are ■paralhl to
C,0„ are

+T, COS. a„ +T, COS. ".., — P, COS. 6„ —"VV", cos. >,
those pressures being taken positively which tend to move
the axis of the drum from 0, towards C,, and those nega-
tively whose tendency is in the opposite direction.

In like manner the pressures resolved perpendicular to
C,C, are

— T, sin. B„ +T, sin. a„ +P, sin. a„ — W, sin. >,
those pressures being taken negatively whose tendency
when tnns resolved perpendicular to OiO^ is to bring that
line nearer to a vertical airection, and those jHmtmelj/ whose
tendency is in the opp<ffiite direction.

Observing that E, is the resultant of all these pressures,
we have (Art. 11.)

E,==KTi+T,)cos. a_P, COB. &~y^, cos. .}'-!-
{P, sin. fl,-(T,-T,) sin. « -W, sin. .{ '.
Proceeding similarly in respect to the pressm^es applied to
the upper drum, we shall obtain

R,'=|(T,+T,) cos. "-P, COS. ^+W, COS. >{'+
jP, sin. i)^+(T,— T,) Bin. «,-W, sin. ij= ;
or substitutmg 2T for T,+T„ and 3i for T,~T,

E,'= jSTcos. a,— P, COS. ^— Wj COS. i','+ ^
jP, sin, flj— 2isin. a,— WjSin. ij'

B,'= JSTCOB. ^— P, COB. fl,+W. COS. i|' +

}P, Bin. ^5+2* sin. ",— W, sin. i| '

. . (219).

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THE BAND.

223

By eliminating li,, B,, and t between the fovir eijiiations
(218) and (219), a relation is determined between the three
quantities P„ P„ T. To simplify this elimination let us sup-
pose that the preceding hypothesis in i-espect to the direc-
tions in which the pressures are to be ii&.(3i positively and
negati^ieby is so made, that the expressions enclosed within
the brackets in the above equations (219) and squared may,
each of tJxem, represent a positive quantity. Let us, more-
over, suppose \hB first of the two quantities squared in each
equation to be considerably sreator than the second, or the
pressure upon the axis of each drum in the direction of the
line 0, 0, joining their centi-es, greatly to exceed the pres-
sure upon it in a direction perpendicular to that line ; an
hypothesis which will in every practical case be realised.
Tnese suppositions being made, we obtain, with a sufficient
degree of approximation, by Poncelet's Theorem*,

E,=al3T COB. a,-P, COS. 0,-W, CM. i} +
|3 5P, sin. fl,-2( ein. ^-W, sin. .} ,

R,=:ct)2T COS. ^— P, COS. e^ + W, COS. i] +

/3|P, sin. e, + 2i sin. «,— W, sin. i\.

Substituting these values of K^ and Ej in equation (318),
and reducing, we have

P,w,— 2*(r,— ^P, sin. ^ sin. ?)=
pJ2«T COS. ^-P^^ -W,y,} am.
P,a,— 2i(r, — ^p, sin, a., sin. (p)=
~»,12" T cos. «,-PA+'W",rJ sin. <f J

where /3,=(tt cos. S,-^ sin. »,),
/3,=(a COS. 9,-/3 sin. flj,
y,=:(a COS. ' + /3 sin, i),
y^-={a. COS. I— /3 sin. i).

Eliminating t between these equations, and neglecting
terms above 5ie iirat dimension in p, sin. ip and p, sin, tp,

( +Pia,(»'j— /3p5 sin. a, sin. p) ) _
I —P 3(15(7',— /3p J sin, a, sin. ip) f ~

j +p,r,(2-T COS. - -P./3,--«r,r,) ) . ,„,

a, being for the most part exceedingly small, the terms

■ (220),

, Google

^pj sin. K, ain. (p and fSp^ sin, a, sin, ip may be noglected; we
shall tlien obtain l>y transposition and reduction

a

+ 2«T(p,j',+p/,) COS.

— {Wjp ,y,r J — W jp ,-YsT,)

. (222).

K this equation be compared witli equation (214), it ■will
be found to agree with it, mutatis mutandds, except that
the co-efficient 2« is in that equation 2. Tliia difference
manifestly results from the apjtrommate character of the
theoram of Poncelet.

Substituting the latter co-efficieut for the former, multiply-
ing both sides of the equation by (1 — ^'sin. tp), neglecting

tenns of more than two d'
reducing.

— , — , and sin. 9, and

which is the relation between the moving and working
pressures in the state bordering npon motion. From this
relation we obtain for the modulus of the band (equation
121)

s.{i%:;+Atsl »"■'•■■«■

If the angle 5, be eoneeiyed to increase until it exceed
n, Pj will pass to the opposite side of C,C„ and /3^ will
become negative; whence it is apparent, that equation

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THE BAND. 923

(234) agrees witli equation (314) in other respects, and in
tlie condition of tiie amtiguous aign. It is moreover appa-
rent, from the foi-ra. assumed by the modulus in this ease
and in that of the preceding article, thxxt the areatest
economy of povier h obtained by amlymg the moving amd
Ihf- wofkmg pressures on the same side of the Une 0,0, joim-
ing the wees of the drums. This is in fact hut a particular
case of the general principle estahlisSied ia Art. 168.

195. The initial tension T of the hand may he deter-
niiied precisely as in the former case (equation 217).
Bepreseiiting by 9 the angle sub-
tended by the circumference which
the band emhracee on the second
or driven drum, by P the maxi-
mum resistance opposed to its mo-
tion at the distance a, by * the
limiting angle of resistance between
tlie band and the siu'face of the
drum, and hy i, and t, the tensions
upon the two pai^ of the band,
when its maximum resistance being opposed, it is uj)on the
point of slipping ; observing, moreover, that in this ease

2{t^—t^) or 2t is represented {Art. 193.) by 2T^ g~ ^ ; then

e +' 1
substituting in the second of equations (220) this value for
2i, and P and a for P, and a„ and neglecting the exceed-
iagly small tenn which involves the product sin. a, sin. ?,
we nave

-^j2aTcos.«,-P/3,+"W",r,Ssin 9.

Also, since a^ represents the inclination of the two partsof
the band to one another ; since, moreover, these touch th&
surfaces of the drams, and that 0 represents the inclination
of the radii drawn from the centre of the lesser drum to the-
toiiching points, therefore 6=-7t— cCj. Substituting this value
of 6 in the above equation, and solving it in respect to T, we
have

15

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Fla-fA ™- »)+"y.p.r. am- V , ,

196. The modrulits of the band when the two parts of it,
which interoene hetweeti the drums, a/re made to cross one
another.

If the directions of the two parts of the band be made
to cross, as shown in the accompanying
figure, the moving pressure T, upon the
second drum is applied to it rn the side
opposite to that on which it is apphed
when the bands do not cross ; so that in
this case, in order that the greatest eco-
nomy of power may be attained (Art.
168.), the working pressure or resistance
P, should be apphed to it on the side
opposite to that in whieh it was apphed
in the otlier case, and therefore on the side of the line C,0„
opposite to that on which the moving pressure P, upon the
first drum is applied. This disposition of the moYing and
working pressures being supposed, and this case being mves-
tigated by the same steps as the preceding, we shall arrive
at precisely the same erpressions (equations 223 and 224)
for the relation of the moving and the working pressures,
and for the modulus.

In estimating the value of the imMial tension T (equation
225) it will, however, be found, that the angle 3, subtended
at the centre 0, of the second drum by the arc KML, which
is embraced by the bond, is no longer in this case repre-
sented by T— a, but by w+k,. This will be evident if we
consider that the four angles of the quadrilateral figure
Cl^IL being equal to four right angles, and its angles at K
and L being right angles, the remaining angles KIL and
KCJj are equal to two right angles, so that KOjL=*—a, ;
but the angle subtended oy KML equals 2*— KC5L; it
equals therefore ir+«i. If this value be substituted for *— a,
in equation (225) it becomes

./Google

TEE TEETn OF

P(m— PjiS, sin. ip)+AV,p, sin. ipy'
-p^ttcos. «iSm. 9

Now tlie fraction in tlie denominator of this expression
being essentially greater in value tlian that in the denomi-
nator of the preceding (equation 225), it follows that the
initial tension T, which must be given to the band in order
that it may transmit the work ftom the one drum to the
other under a given resistance P, is less when the two parts
of the hand cross than when they do not, and, -therefore, that
the modulus (equation 23i) is less; so that the hand is
worked with the greatest economy of power {otJier things
Imng the same) when the two parts of it which intervene
be6ioeen the drums cure made to cross one another. Indeed it
is evident, that since in this case the arc; embraced by the
band on each dram subteni^ a greater angle than in the
otlier case, a less tension of the band ia this case than in the
other is required (Art. 185.) to prevent it from slipping
under a given resistance, bo that the friction upon the axis
of the di'ums which results from the tension of the band is
less in this case than the other, and therefore the work
expended on that friction less iu the same proportion.

The Teeth of Wheels.

197, let A, B represent two circles in contact at D, and
moveable about fixed centres at 0, and C,. It
ia evident that if by reason of the friction of
these two circles upon one another at D any
motion of rotation ^ven to A be communicated
to B, the angles PD,D and QOJ) described in
the same time by these two circles, will be such
as will make the arcs PD and QD which they
subtend at the circumferences of the circles equal to one
another. Let the angle PC,D* be represented by ^i, and the
angle QO^D by 6^; also let the radii 0,D and C,!) of the cir-
cles be represented by r, and r,. Now, arc PD=7'ifl„ arc
QD=r,fl, ; and since ?D=QD, therefore r^K=r^S^ ;

* Or rather the are which this angle subtend? to radius iinitj.

, Google

THE TEETn OF WHEl!;LS.

■■■t = ir {«).

The angles described, in the same time, by two circles
which revolve in contact are therefore inrei-sely proportio)>ai
to the radii of the circles, so that their angular velocities
(Art. 74.) bear a constant proportion to one another ; and if
one revolves imifonnly, then the other revolves unifoiToly ;
if the angular revolution of the one varies in any proportion,
then that of the other varies in like proportion.

"When the resist(mce opposed to the rotation of the driven
circle or wheel B is considerable, it is no longer possible to
give motion to that circle by the fiiction on its circum-
leience ol the dn-\mg circle. It becomes therefore neces-
sary m the gieat majority of cases to cause the rotation of
the diiven wheel by some other means than the friction of
the ciicomleieace ot the diiving wheel.

One expedient is the band already described, by means of
which the weels may be made to drive one another at any
distances of thetr centre'*, and under a far g-eater resistance
than they could by then mutual contact. \Vhen, however,
the ptessme is considerable, and the wheels may be brought
mto actual contact, the common and the more certain
method is to transfer the motion
ftom one to the other by means of
piojections on the one wheel called
TEETH, which engage in similar pro-
jections on the other.

In the construction of these teeth
the problem to be solved is, to give
such shapes to their surfaces of mu-
tual contact, as that the wheels shall
be made to tnm by the intervention
of their teeth prtci^elj a^ they would by the friction of
then circumferences

(i^'Yf?^h

•^S-*:

IS** Tiiat it w jtossiilp to construct teeth which shall
answer this condition may thus be shown.
Let mm and m'n' be two curves, the one
described on the plane of the circle A, and
the other on the plane of the circle B ; and
let them be such that as the circle A re-
1 volves, carrying round with it the circle B,
by their mntual contact at D, these two
curves mil and m'n' may continually touch

./Google

THE TEETH OF WHEELS,

one a?tother, altering of course, as they will do continually,
their relative petitions and their point of contact T.

It ia evident tliat the two circles -would be made to
revolve ty the contact of teeth whcse edges were of the
forms of these two cui'vee mw and m'n' precisely as tliey
would by their friction upon the circumferences of one
another at the point D ; for in the former case a certain
series of points of contact of the circles {infinitely near to
one another") at D, brings about another given series of points
of contact (infinitely near to one another) of the curves mn
and m'n' at T ; and in the latter case the same series of
points in the cui'ves mn and m'n' brought into contact neces-
sarily produces the contact of the same series of points in
the two circumferences of the two eircles at J).

To construct teeth whose surfaces of contact shall p
the properties here assigned to the curves t/m and m'n
the problem to be solved. Of the solution of this problem
the following is the fundamental principle :

199. In order that two circles A and E may he made to
r&volve by the contact of the surfaces irm and m'n' of their
te^h, preoisel/y as they would oy the friction of their tdr-
(Mwferenoes, it is necessary, and it is suf
fieient, thai a line dra/uyn from, the point
of contact T! of the teeth to the point of
contact D of the d/rciM-nf&tences s/iould, in
\ mery position of the point T, he p&rp&ndi-
' cular to the swjacea in contact there, i. e.,
a normal to hoth the curves mn and m'n'.

To prove this principle, we must first ^tablish the foUow-
t; — If two circles JI and N be made to revolve
about the fixed centi-es E and F by their mu-
tual contact at L, and if the planes of these
circles be conceived to be earned round with
\ them in this revolution, and a point P on the
' plane of M to trace out a curve PQ on the
plane of W whilst thus revolving, then is this
curved line FQ precisely the same as would
have been described on the plane of N by the same point P,
if the latter plane, instead of revolving, had remained at
rest, and the centre E of the circle M having been released

./Google

230 THE TECTH OF WHEELS.

froin its axis, tliat circle Itad been made to roll (carrying its
plane with it) on the circnmference of N. For conceive 0
to represent a third plane on wliich the centres of E and F
are iixed. It is evident that if, whilst the circles M and N"
are revolving by their mutual contact, the plane 0, to which
then' centres are both fixed, be in any way moved, no change
will thereby be produced in form of tlie curve I'Q, which the
point P in the plane of M is desci-ibing upon the plane of N,
euch a motion being eofmnon to both the planes m. and N.*
]Nrow let the direction in which the circle N is revolving be
that shown by the arrow, and its angular velocity uniform ;
and conceive the plane 0 to be made to revolve about F with
an angular velocity (Art. 74) which is eq^ual to that of N,
but in an opposite direction, communicating
this angular velocity to M and If, these re-
volving meantime in respect to ono another,
) and by their mutual contact, precisely as tliey
' did before.!

It is clear that the circle N being carried
round by its own proper motion in one direc-
tion, and by the motion common to it and the plane 0 with
the same angular velocity in the opposite direction, will, in
reality rest in space ; whilst the centre E of the chcle M,
having no motion proper to itself, will revolve with the
angular velocity of ttie plane O, and the various other points
in that circle with angular velocities, compounded of their
proper velocities, and those which they receive in common
with the plane 0, these velocities neutralising one another
at the pomt L of the circle, by which point it is in contact
with the circle N. So that whilst M revolves round N, the
point L, by which the former circle at any time touches the
other, is at rest ; this caulescent point of the circle M never-
theless continually varying its position on the circumferences
of both circles, and the circle il being in fact made to roll
on the circle N at rest.

Tlius, then, it appears, that by communicating a certain
common angular velocity to both the circles M. and N about

* Thus for inetance, jf the circles M and N contioue to revolve, wa i
eTittentlj place the whole machine in a ship under sail, in a mOTlng oarrit
or upon a revoMng whee!, witbont in the leait altering the form of the ou!
whidi the point P, ceTolving with the plane of the drcle M, ia made to ti
on the plane of N, because the motion wa have ooinmunieated is commoi
both theea circles.

f M Bod N may be imagined V> be placed upon a horizontal wheel 0, firs
rest, and then made to revolve bachxards in respect to the motion of N.

, Google

THE TEETH OF WHEELS. 231

the centre F, tlie former eii'cle is made to roll iipoii the othei
at rest ; and, moreoTer, that this common angular velocity
does not alter the form of the curve PQ, which a point P in
the plane of the one circle is made to trace upon the plane
of the other, or, in other words, that the curve traced under
these circumstances is the same, whether the circles revolve
round fixed centres by their mutual contact, or whether the
centre of one circle be released, and it be made to roll upon
the circnraierence of the other at rest.

This lemma being established, the truth of the proposition
stated at the head of this article becomes evident ; for if M
roU on the circumference of N, it is evident that P will, at
any instant, be describing a circle about their point of con-
tact L.*

Since then P is describing, at every instant, a cii'cle about
L when M rolls upon N, N being fixed, and since the curve
described by P upon tins supposition is precisely the same
as would have been traced by it if the centres of both cir-
cles had been fixed, and they had turned by their mutual
contact, it follows that in this last case (when the circles
revolve about fixed centres by their mutual contact) the
point P is at any instant of the revolution describing, during
that instant, an exceedingly small circular arc about the
point L ; whence it follows mat PL is always a perprndicu-
tar to the curve PQ at the point P, or a iwrmal to it.
!N^ow let _p be a point exceedingly neai' to T in the curve
TOft', which curve is fixed upon the plane
of the circle A. It is evident that, as the
point p passes through its contact with the
curve mn at T (see Art. 198.), it will be
I made to describe, on the plane of the circle
I B, an exceedingly small portion of that
cm've mn. But the curve which it is
(under tliese circumstances) at any instant
describing upon the plane of B has been shown to be
always pei-pendicular to the line DT ; tlie curve mn is there-
fore at the point T pei'pendicular to the hne DT ; whence it
follows that tlie curve m'n' is also perpendicular to that line,
and that DT is a normal to hoik those. <yu/rves at T. This ia
the characteristic property of the curves mn and m'n', so that
they may satisfy tlie condition of a continual contact with

* For either circle maj be imagined to be a polygon of an infinite number
of sides, on one of the ajiglca of ivhioli tha rolling cii'Olc will, at anj iustant,
be in tlie act of turning.

, Google

232 THE TKETH OF WHEELS.

one another, whilst the eireles revolve bj tlie contact of
their circumference at D, and therefore conversely, so that
these curves uiay, by their mutual contact, give to the cii'-
clee the same motion as they would receive from tlie contact
of their circunrferencea.

iOO. To disoriie, hy "means of oiroular a/rcs, the form of a
tooth on mie wheel which shall work truly with a tooth of
any gw&nform on anothm- wheel.

Let the wheels be required to revolve by the action of
their teeth, as they would by the
contact of the circle ABE and
ADF, called Sh^xr ^mitwe ov pitch
circles. Let AB represent an ai-c
of tlie pitch circle ABE, included
between any two mnilar points A
and B of consecutive teeth, and let
AD represent an are of the pitch
circle ADF equal to the arc AB, so
that the points D and B may come
simultaneously to A, when the cir-
cles are made to revolve by tlieir
mutual contact. AB and Al> are
called the pitches of the teeth of the two wheels. Divide
each of these pitches intfl the same number of equal parts
in the points a, h, &e., a', V, &c. ; the points a and a', t and
V, &c,, will then be brought simuUcmeously to the point A.
Let Tim represent the form of the face of a tooth on the
wheel, whose centre is C„ with which tootli a corresponding
tooth on the other wheel is to wort tmly ; that is to say,
the tooth on the other wheel, whose centre is C^, is to be cut,
so that, driving the surface «wi, or being driven by it, the
wheels shall revolve precisely as they would by the con-
tact of their pitch circles ABE and ADF at A. From A
measure the least distance Aa to the curve mn, and with
radius Aa and centime A describe a cu-cular ai-c k/3 on the
plane of the circle whose centre is 0,. From a measure, in
like maimer, the least distance ao.', to the curve mn, fmd
with this distance a^' and the centre a, describe a circular
arc ^Y, intersecting the axe «/3 in /3. From the point b
measure similarly the shortest distance W to mn, and with

./Google

the centre V and this distance h^f describe a circular ai'C
-/(!, intersecting ^7 in 7, and so with the other points of
division. A curve touching these cii'cular arcs t^, /S/, yS^
&c., ivill give the ti'ue surface or boundary of the tooth.*

In order to prove this let it be ohserved, that the shortest
distance a^' from a given point a to a given curve mn is a
nonnal to the cxu've at the point of in which it meets it ; and
tlierefore, that if a circle be struck from tliia point a witli this
least distance as a radius, then this circle mnst touch the
curve in the point a-', and the cui-ve and circle have a com-
mon normal m that point.

Now the points a and a' will be brought by the revolution
of the pitch circles simultaneously to the point of contact A,
and the least distance of the curve mn from the point A will
then he aa', so that the arc /3y will then be an arc struck
from the centre A, with this last distance for its radius. This
circular arc /Sy will therefore touch the curve mra in the point
0.' and the line a^--', which will then be a line drawn ti-om
the point of contact A of the two pitch cirelee to the point
of contact a' of the two curves inm, and mV, will also be a
normal to both curves at that point. The circles will there-
fore at that instant drive one another (J Art. 196.) by the con-
tact of the surtaces m/n and m!n', precisely as they would by
the contact of their circumferences. And as every circular
are of the curve m/n' similar to 0y becomes in its tnm tlie
acting surface of the tooth, it will, in like manner, at one
fomt work truly with a corresponding point of mn, so that
the circles will thus drive one anotlier tml^ at as many
points of the surfaces of their teeth, as tliere have been taken
points of division a, J, &c. and arcs a/3, ^y. &c.t

the number of these pointe of division, the more aecui'ate the
form of the tooiJu It appears, howeyer, to be sufBoiont
in most CHaes, to talte three points of diTJsioa, or even
two, where no great accuracy is required. M. Poacelet
{Mk. Indmi. 8""^ parUe, Art, fiO.) has given the following,
yet ea^er, method by which the true form of the tooUi
may be ^roximate^ to with suffident aocumcy in most
caaea. Suppose the given tooth N upon iJie one wheel to
be placed in the posltiou in which it ia first to engage or
disengage from the reqmred tootli on the other wheeL
and let Aa and AS be equal arcs of ^le pitch circles of

'3:- the two wheels whose point of contact is A. Draw Aa

the shortest distance hetweenA and the face of the tooth

N; join Qa; bisect that line in m, and draw miv perpeadi-

ouiar to aa intersecting the oircuioforenoe Ao in n. If

[ from the centre n a circular arc be described passing

J through the points a and a, it will ^ve the required fonn

of tlie tooth nearly,

, Google

IMTOLUTE TEETH.

IrrvoLxriE Teeth.

201. The teeth of two wheels wUl work trvk/ together if tli^
he hounded hy curves of the form traced out l/y the eietremity
of a Jlesaile Une, wi/windvng from the droumferenes of a
circle, <md called the im)olute of a evrde,provided that the
circles of which these are the moolutes be concentric with
the pitch mrdea of the wheels, and hm>e tfmr radii in the
same jjrojmtion with the radk of the pitch ci/rcles.

Let OE and OF represent tiie pitch circles of two "wheola,
AG and EH two circles eonceutric witli
them and having their radii 0,A and 0,E
ill the same proportion with the radii OiO
and OjO of the pitch circles. Also let mn
and rrh'n' represent the edges of teeth on the
two wheels struck by the extremities of ilexi-
hie lines unwinding from the circamferencea
, of the circles AG and BH respectively. Let
I these teeth be in contact, in any position
of the wheels, in the point T, and from the
point T draw TA and TB tangents to the
generating circles GA and BH m the pointj,
A and B. Then does AT evidently represent the position of
the flexible line when its extremity was in the act of gene-
rating the point T in the curve mn; whence it follows, that
AT ia a normal to the curve rnn at the point T* ; and in
like manner that BT is a normal to the curve m/n' at the
same point T. Now the two curves have a
common tangent at T ; therefore their nor-
mals TA and TB at that point are in the same
straightline,b6ingboth perpendicular to their
tangent there. Since then ATB ia a straight
line, and that the vertical angles at the point
0 where AB and C,Oj intersect are equal, as
\ also the right angles at A and B, it follows
i that the triangles AoO,and BoC, are similar,
and that C,o : G,o :: O.A : C,B. But 0,A :
C,B :: 0,0 : 0,0; .-. 0,o : C,o : : 0,0.
C„0 ; therefore the points O and o coincide,
and the straight line AB, wliich passes throiigh the point of

• For if the circle be conoeiyed a polygon of an infinite number of eidea, It
!s evident that the line, nhsn In the act of unnlnding from it at A, is tuining
upon one of the angles of that polygon, and therefore that its eitremjty is,
through an inflnitolj sninll angle, descvlbing a circular ivro about that point.

, Google

INVOLUTE TEETH.

235

contact T of the two teetli, and is porpendicular to the eur-
faces of both at that point, pasBta also through the point of
contact 0 of the pitch circles of the wheels. Now this ia
true, whatever bo the positions of the wheels, and wliatever,.
therefore, be the points of contact of the teeth. This then
the condition established in Art. 199, as that necessary and
sufficient to the true action of the teeth of wheels, viz, " that
a line drawn from the point of contact to the pitch circles to
the point of contact of tho teeth should be a normal to their
surfaces at that point, in all the different petitions of the
teeth," obtains in regard to involute teeth.*

The point of contact T of the teeth ■moves along the straight
!ine AB, which ia drawB touching the generating circles BH
and AG of the involutes ; this Hue is what is called the lociis
of the different points of contact. Moreover, thia property
obtains, whatever may be the number of teeth in contact at
once, ao that all the points of contact of the teeth, if there
be more than one tooth in contact at once, lie always in this

line ; which ia a characteristic, and a most important pro-
perty of teeth of the uivolute form. Thus in the above

* The author proposes the following illustration of the action of involute
teeth, which he believes to be neiv. Coiioeive AB to represent a band passing
round the clrelea AG and BH, the wheels would evidently be driven hy this
band precisely fis thej would by the contact of their pitch oiroleB, einoe the
radii of AG and BH are to one another bf? the radii of the pitch dniee. Gon-
eeive, moreover, that the eirclea BH and AG nnwj round with them their
planei ss they revolve, and that a tracer ie fixed at any point T of the band,
tradng, at the fiame time, linfes mn and mV, upon both plania, as theycevolve
beneath it. It is evident that thcae curves, being traced by the earae point,
must be in contact in all portions of the cicclcB when driven by the band, and
therefore when driven by their mutu^ contact. The wheels would therefore
be driven by the contact of ieeth of the forms run and in'n' thus traced by the
point T of the band pceciBcly as they would by the contact of their pitch cir-
cles. Now it is ea^ly seen, that the curves mra and ra'«', thus described by the
point T of the band, ace imohdes of the circles AG and BH.

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2«6 EPICYOLOIDAL AND IIYrCCYCI.OIDAL TEETH.

figure, which represents part of two wheels wifli involute
teeth, it will be seen that the points r s of contact of the
teetli aro in the same straight hne touching the haee* of one
ol' the involutes, and passing through the point of contact A
of the pitch circles, as also the points A and I in that touch-
ing the base of the other.

EpicTCLomAL AHD Htpoctcloidai. Tbeth.

202. If one circle be made to roll externally on the cir-
cumference of another, and if, whilst this mo-
tion is taking place, a point in the circumfe-
rence of the rolling eh'cle he made to trace
out a cmwe upon the plane of the fixed circle,
tlie curve so generated is called an EPicTCLorD,
the rolling circle being called the mneratmg
\ circle of the epicycloid, and the circle upon
] which ifr rolls its base.

If the generating circle, instead of rolling
on the outside or convex circumference of its
e, toII on its inside or concave ckcumfe-
rence, the curve generated is called the hypoctcloid.

Let PQ and PK be respectively an epicycloid and a hypo-
cycloid, having the same generating circle APH, and
having for their bases the pitch circles AP and AE of two
wheels. If teetli be cut upon these wheels, whose edges
coincide with, the curves PQ and PE, they will work truly
with one another ; for let them be in contact at P, and let
their common generating cii-cle APH be placed bo as to
touch the pitch circles of both wheels at A, tlien will its cir-
cumference evidently pass through tlie point of contact P
of the teeth : for if it be made to roll through an excooil-
ingly small angle upon the point A, rolling there upon the
circumference of ooth circles, its generating point will
traverse exceedingly small portions of both curves ; since
then a given point in the circumference of the circle APH
is thus shown to be at one and the same time in the perime-
ters of both the curves PQ and PE, that point must of
necessity be the point of contact P of the curves ; since,

• The circles from which the involutes are deaoribed are called their bases.
This cnt and that at page 231. are copied from Mr. HawkiuE' editiou of Camug
an the Teeth of Wheels.

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EPICTCLOIDAL AND HYPOCYCLOIDAJj TEETH. 237

moi-eoTer, when the circle APH rolls upon tlio point A, its
generating point t/raroerses a small portion of the perimeter
of each of the curves PQ and PR at P, it follows that the
line AP is a normal to both cnrvea at that point ; for whilst
the circle APH is rolling through an exceedingly small
angle upon A, the point P in it, is describing a circle about
fJiat pomt whose radius is AP.* Teeth, therefore, whose
edges are of the forms PQ and PK satisfy the condition
that the line AP drawn from the point of coatact of the
pitch circles to any point of contact of the teeth ia a normal
to the SYirfac^ of both at that point, which condition has been
shown (Art.. 199.) to be that necessary and snfficient to the
correct working of the teeth.f
Thus then it appeare, that if an epiovdoid be desciibed

■ Tha circle APH may be conddered a polygon of an infiniie namber of
rides, ott one of the angles of which polygon it may at any instant be con-
c«LVed to be tramiiig.

+ The entire demonstration by which it has been here ehown that Che
curves generated by a point id the oiroumferenoe of a giTeil generating circle
APH rtSling upon tlie convex oireumferenee of one of the pitch dreies, and
upon the oonoave ciroumferenoe of the other are proper to form the edges of
contact of the teeth, is eTideiitly applicable if any other generaOng curve be
Bubstituted for APH. It may be shown preoiaely in the same manner, that
the curtes P(J and TR generated by the rolling of any suoh curve (not being
a Mccle) upon the pitch circles, possess this property, that the line PA drawn
from any point of their contact to the point of contact of their pitch circlea
is a nonnai to both, which property ia necessary and sufficient to their correcS
aedon as teetJi. This was first demonstrated aa a general principle of the con-
struciJon of the teetli of wheels by Mr. Airy, in tlie Cambridge Phil. Trans,
vol. ii. He has farther shown, tliat a tooth of any form whatever being out
upon a wheel, it is posable to find a curve which, rolling upon the patch circle
of that wiieei, ahali by a certain generating point- traverse the edge of the
^ven tootli. The curve tliue found being made to roll on the oironmfereuca
of the pitch circle of a second wheel, will therefore trace out the form of a
tooth which will work truly with the first. This beaatiful property involves

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238 EPIOYOrXUDAL AND HTPOOYOLOIDAL TBETET.

on the plane of one of the wheels -with any geterating
circle, and with the pitch circle of that wheel for its hase ;
and if a hypocycloid be described on the plane of the other
wheel with the pitch circle of ^Aa* wheel for its base; and
if, the faces or acting siu'faces of the teeth on the two
weeels he cut so ae to coincide with this epicycloid and this
hypocycloid respectively, tlien will the wheels be driven
correctly by the intervention of these teeth. Parts of two
wlieels havmg epicyeloidal teeth are represented in the pre-
ceding figure.

i03. Lemma. — If the diameter of the generatmg circls of a
hypocycloid equal the radius of its hose, the hyponuAoid
iecomes a straight line having the direction of a radvus of
its hose.

Let D and d represent two positions of the centre of such
a generating circle, and suppose the
generating point to have been at A in
tlie first position, and join AC ; then
will the generating poijit be at P in the
second position, i. e. at the point where
CA intersecte the- circle in its second
position; for join Co and Vd, tlien
_ /Ptfa=/PCi+ZCFf;=.2ACg. _A.l60
2(?ffl— OA ; .-. 2da x Pda=20AxA.Ca ; :. daxVda^CA X
ACa; .".arc Aa=arc Ta. Since then the arc aP equals
the arc aA, the point P is that which in the first position
coincided with A, «. e. P ie tlie generating point ; and tliis
is true for aU positions of the generating circle ; the gene-
rating point is therefore always' in the straight line AC.
The edge, tlierefore, of a hypocycloidal tooth, the diameter
of whose generating circle eqnals half the diameter of the
pitch cii'cle of its wheel, is a straight line whose direction
is towards the centre of the wlieel.*

the theoretical solution of the proMem which Poncekt has solved by the
geometrical oonstruetion given to Ai'tiole 300. If ilie rolling curve be a
logftrithmio flirfral, the involnte form of tooth will be generated.

" The following very ingenious HppUcfition haa been made of tliie property
of tlie hypooycloid to conTert a circular inlo an alternate rectilinear motion.
AB repreeenta a ring of metal, fixed in position, and having teeth cut upon itf

, Google

TO BET OUT TilE TKETK OF WHEELS.

To SET OUT THE TeETH OS WhBELS.

20i. All the teeth of the same wheel are constructed of
the eame form and of equal dhnensions ; it would, indeed,
evidently be impossible to coustrnct two wheels vdtii dit-
ferent numbei-s of teetli, which should work truly with one
another, if all tlie teeth on each wheel were not thus alike.

All the teeth of a wheel are therefore set out by tlie work-
man from the same pattern or model, and it is in determining
the form and dimensions of this single pattern or model of
one or more teeth in reference to the mechanical effects
which the wheel is to produce, when all its teeth are cut out
upon it and it receives its proper place in the mechanical
combination of which it is to form a part, that consists the
art of the description of the teeth of wheels.

The mechanical function usnallj' assigned to toothed wheela
is the transmission of work under an increased or diminished
velocity. If CD, DE, &c., represent ai'cs of the pitch circle

oonoaTa oifoumference. C ia the centre of
a wheel, baying teeth out in its circum-
ference to work ivith those upon the dreurn-
ference of the ring, tind ha.viug the diame-
ter of its pitch circle equal to half that of
tlie pitch circle of the teeth of the ring.
This being the caae, it is evident, tiiat if the
pitch circle of the wheel 0 were made to
roll upon that of the ring, an; point in its
circumference would deeoribe a straight line
passing throueh the centre D of the ring ;
but the circle C would roll upon the ring by
the mutual action of thdr teeth as it would
by the contact of their pitch dcclee ; if the
circle C then be made to roll upon the ring
by the interrention of teeth out upon both, any pomt in the oiicumfereocB ol
C will describe a straight line paasing through D. Now, conceive C to be thiis
made to roll round the ring Ijy meana of a double or forked link CD, between
the two branches of whii^ the wheel is received, beu^ perforated at their
extremities by ctteular apertores, which aerve as bewhiga to the aohd aris of
the wheeL At its other eifremity D, thie forked link is rigicl\j connected
with an asis passing through the centre of the ring, to which axis is commu-
nicated the circnlar motion to be converted by the inatrnmant into an- altei'-
naUng rectilineal motion. This circular motion will thus be made to carry
the centre C of the wheel ronnd the pomt D, and at the same time, cauae it to
roll upon the circumference of the ring. Now, conceive the axis 0 of the
wheel, which forms part of the wheel itself, to be proluiiged beyond the collar
in which it turns, and to have ri^dly fixed upon Its eitremity a bar OP. It is
evident that a point F in tliia bu', whose distance from the axis C of the wheel
equals the radius of its pitch circle, will move precisely as a point in the pitch drole
of the wheel moves, and therefore that it will describe continually a straight
hue passing through the centre D of the ring. This point P cacaivea, there
fbre, the alternating rectilinear motion which it was required to communicate.

, Google

r THE TEETH OF WHEELS.

of a wheel intercepted "between similar points of consecutive
teeth {the chords of which arcs are called the pitches of the
teeth), it is evident that all these arcs mnet be equal, since
the teeth are all equal and similarly placed ; so that each
tooth of either wheel, aa it passes through its contact with a
corresponding tooth of the other, cai-ries its pitch line through
the same space CD, over the point of contact C of the pitch
lines. Since, therefore, the pitch line of the one wheel is
carried over a space equal to CD, and that of the other over
a space equal to cd by the contact of any two of their teeth,
and since the wheels revolve by the contact of their teeth
as they would by the contact of their pitch circles at C, it
follows tliat the arcs CD and od are equal. Now let r, and
?■, represent the radii of the pitch circles of the two wheels,
then will 2w, and StiTj represent the circumferences of their
pitch circles; and if n, and n^ represent the numbers of

teeth cut on tliem respectively, then CD= — ■' and cd^ — ',
■^ '' til %

theretore, j

■ (227);

Therefore the radii of Hie pitch circles of the two wheela
must be to one anotlier as mo numbers of teeth to be cut
upon them respectively.

Again, let m^ represent the number of revolutions made
by the first wheel, whilst m, revolutions are made by the
second ; then wiU SriTim, represent the space described by

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-A. TKAIS OF WHEELS,

341

tile circumference of the piteli circle of the first ■wheol while
these revolutions are made, and 2Trr,m, that deacribed by the
circmnference of the pitch circle of the second ; but the
wheels revolve as though their pitch circlcB were in contact,
therefore the circumferences of these eifcles rerolye tlirough
equal spaces, therefore 27r7',mi,=27ir,?«j ;

The radii of the pitch circles of the wheels are therefore
iiivei-sely as the numbers of revolutions made in the same
time by them.
Equating the second members of equations (227) and (328)

Tlie numbers of revolutions made by the wheels in the same
time are therefore to one another invci'sely as the uumbei-iii
of teeth.

205. Jh, a tram of wheels, to deterTinne how "many revolutions
the last wheel makes whilst ths jhst is m,aMng imy given

When a wheel, driven by anotlier, candea its axis round
with it, on which axis a third
wheel is fixed, engaging with and.
giving motion to s. fourth, which,
m like manner, is fixed upon ite
axis, and carries round with it a
fifth wheel fixed upon the same
axis, which fifth wheel engages
with a sixth upon another axis,
and so on as shown in the above figure, the combination
forms a trim^ of wheels. Let «.„ n,, «.„... n,p represent tlie
numbers of teeth in the successive wheels forming such a
train of J? pairs of wheels ; and whilst the firet wheel ia
making m revolutions, let the second and third (which revolve
togetlier, being fixed on the same axis) make m, revolutions ;
the fourth and fifth (which, in like manner, revolve together)
OT, revolutions, the sixth and seventh ot,, and so on ; and let
the last or tp^ wheel thus he made to revolve »%, times whilst
16

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242 A TKAIN OF WHEELS;

tlie first revolves wi times. Then, since tlie first wheel wliich
has w, teeth gives motion to the second which has n, teeth,
and that whilst the former makes "m revolutions the latter

makes m, revolutions, therefore {equation 229), ^ = — ;

and since, while the tlurd wheel (which revolves with the
second, makes m^ revolutions, the fourth makes «i, revolu-
tions ; therefore, — ° = — . Similarly, since while tlie fifth
wheel, which has n^ teeth, makes m^ revolutions (revolving
with the fourth), the sixth, which has n, teeth, makes m, revo-

lotions ; therefore — = — . In like manner — = — , wc &c.

_^Z_=_^zl. Multiplying these equations together, and
w!p_ 1 nip

striking out factors common to the numeratJjr and denomi-
nator of the first member of the equation which results from
their multiplication, we obtain

lYkp n, . n, . n^ . . . . «ap-i
in ~ n^ . n^ . n^ . . . . n^p

. (230).

The factors in the numerator of this fraction represent the
aiumhers of teeth iu all the driving wheels of this train,
jBid those in the denominator the numbers of teeth in the
driven wheels, or followora as they are more commonly
called.

If the numbers of teeth in the former be all equal and
represented by Ji„ and the numbers of teeth in the latter
also equal and represented by n„ then

= (£)••

. (231).

HaTing determined what should be the number of teeth
in each' of the wheels which enter into any mechanical
combination, with a reference to that particular modification
of the velocity of the revolving parts of the maeliine which
is to be produced by that wheel,* it remains next to consider,
what must he the dimensions of each tooth of the wheel, so

" The reader is referred for a more complete diBouBBion of this subject (which
belongs more pftrUcularlj" to descriptive mechanics) to Professor WiJUs's Frio-
fliploa .of MediHnism, chap. Tii., or to Camua on the Teeth of Wheels, by Haw-

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THG STBESGTH OF TEETH. HiC

that it may be of sufficient etrenstli to transmit the work
which is destined to pass through it, under that velocity, or
to bear the pressure which accompanies the transmission of
that work at that particular velocity ; and it remains fm'ther
to determine, what must be the dimensions of the wheel
itself conseq^nent iipon these dimensions of each tooth, and
this given number of ite teeth.

206. To determine the pitch of the teeth of a wh
the work to he transmitted by the whet

Let U represent the number of units of work to be trans-
mitted by the wheel per minute, m- the number of revolutions
to be made by it per miniite, ti the number of the teeth to
be cut in it, T the pitch of each tooth in feet, P the pr^sure
upon each tooth in pounds.

Therefore nT represents the circumference of the pitch
circle of the wheel, and ttmT represents the space in feet
described by it per minute. Now U represents the work

transmitted by it tWoagh ih.is space per minute, therefore — =i

represents the memi pressure under which this work is trans-
mitted (Art. 50.) ;

The pitch T of the teeth would evidently equal twice the
breadth of each tooth, if the spaces between the teeth were
equal in width to the teeth. In order that the teeth of
wheels which act together may engage with one another and
extricate themselves, with fa«ditj, it ie however necessary
that the pitch should exceed twice the breadth of the tooth
by a quantity which varies according to the accuracy of the
construction of the wheel from -^tk to Ath of the breadth.*

Since the pitch T of the tooth is dependant upon its
breadth, and that the breadth of the tooth is dependant, by
the theory of the strength of materials, upon the pressure P
which it sustains, it is evident that the quantity P in the
above equatioii is a function of T. This ftmctionf may be
assumed of the form

• For a full discusMon of this Eubject see Professor Willis's Princiiiles of
Meehaoism, Arts. 107-112.
t See Appendis, on tlie dimensions of wheels.

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244 TEE STRENGTH OP TEETH,

T=ci/P (233);

where e is a constant dependant for its amount upon the
nature of the material out of ■wliieh the tooth is formed.
Eliminating P hetweeu this equation and the last, and solving
-■ ittoT,

=V'^.

The number of units of work transmitted by any machine

Eer minute is usuaUy represented in horses^ power, one
orse's power being estimated at 33,000 units, so that the
number of horses' power ti-ansmitted by the machine means
tlie number of times 33,000 imits of work are ti-ansmitted by
it every minute, or the number of times 33,000 must be
taken to equal the number of units of work transmitted by
it every minute. If therefore H represent the number of ■
horses' power transmitted by the wheel, then U^33,0O0H.
Substituting this value in the preceding equation, and repre-
Benting the constant 33,000(;' by C, we have

. (234).

The values of the constant 0 for teeth of different mate-
rials are given in the Appendix.

20T. To determme ths radms of the pitch circle of a wheel
■whioh shall contcdn. n teeth of a gvoen pitch.

Let AB represent the pitch T of a tooth,

and let it be supposed to coincide with it«

' chord AMB. Let E represent the radius AC

of the pitch circle, and n the number of teeth

to be cut upon the wheel.

Now there are as many pitches in the cir-
cumference as teetli, therefore the angle AOB

subtended by each pitch is represented by—.

0 T=2AM=2AC"sin. iACB-2K sin. - ;
n

.-.E^^Tcosec- (235).

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TO DESCKtBE ETICYCLOIDAL TEKTH.

)8. To inuks thi patt&i^ of an epieydoidal tooth.

Having determined, asabove,
tlie pitch of the teeth, and the
radius of the pitch circle, strike
an arc of the pitch circle on a
thin piece of oak board or me-
tal plate, and, with a fine saw,
cut the board through along
the circumference of this cir-
cle, so as to divide it into two
parts, one having a convee and
the other a con-esponding oojv-
'^ came circular edge. Lee EF
■ represent one of these portions
of the board, and GH another.

Describe an ai'c of the pitch circle upon a second board oi'
plate from which the pattern is to he cut. Let MK repre-
sent this arc. Fix the piece GH upon this boai-d, so that its
circular ed^e may accurately coincide with the circumference
of the arc MN. Taie, then, a circular plate D of wood or
metal, of the dimensions which it is proposed to give to the
generating circle of tiie epicycloid ; andlet a small point of
steel P be fixed in it, so that this point may project slightly
from its inferior surface, and accurately coincide with its cir-
cumference. Having set off the width AB of the tooth, so
that twice this width increased by from -rVth to tV*^ ^^ *'**''
width (according to the accuracy of workmanship to be
attained) may equal the pitch, cause the circle D to roll upon
the convex edge GK of the board GH, pressing it, at the
same time, slightly upon the surface of the board on which
the arc MN is described, and from which the pattern is to he
cut, having caused the steel point in its circumference first
of all to coincide with the point A ; an epicycloidal arc AP
will thus be described by tlie point P upon the surface MN.
Describe, similarly, an epicycloidal arc BE through the point
B, and let them meet in E.

Let the board GHnow be removed, and let EF be applied
and fixed, so that its concave edge may accurately coincide
with the circular arc MH". "With the same circular plate D
pressed upon the concave edge of EF, and made to roll upon
it, cause me point in its circumference to describe in like
manner, upon the surface of the board from which the pat-
tern is to be cut, a hypoeocloidal arc EH passing through the

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24C

TO DESCRIBE EPICTCUIIDAI, TEETH.

point E, and another AI passing througli the point A. HEl
will then represent the form of a tooth, which will woi'k cor-
rectly (Art. 202.) with the teeth svimlarl/y out upon any other
wheel ; provided that the pitch of the teeth so cut upon the
other wheel be equal to the pitch of the teeth upon this, and
provided that the sanne gm&ratmg cwde D be used to sf/nhe
the Gwrves iipon the two wheels.

209. To detei'mine the prop&r leTigths of epicyoloidal teeth.

The general forms of the teeth of wheels being determined
hy the method explained in the preceding article, it remains
to cut them off of such lengths as may cause tliem succes-
sively to taiie Tip the work from one another, and transmit it
under the circumstances most favourable to the economy of
its trausmisaion, and to the durability of the teeth.

In respect to the economy of the power in its transmission,
it is customary, for reasons to be assigned hereafter, to p^ro-
vide that no tooth of the one wheel should come into action
with a tooth of the other until both are in the act of passing
through the line of centres. This condition may be satisfied
in all cases where the numbers of teetli on neither of the
wheels is exceedingly small, by properly adjusting the
lengths of the teeth, tet two of the teeth of the wheels be
in contact at the point A in the line CD, joining the centres
of the two wheels ; and let the wheel whose centre is 0 be
the driving wheel. Let AH be a portion of the circumfe-
rence of me generating circle of the teeth, then will the
points A and E, where this circle intersects the edges of the

teeth 0 and K of tlie driving wheel, be points of contact

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EPICVCLOIDAL TEETH. 24?

witli tlie edges of tlie toeth M and L of the dTiveu wheel
(Art. 202.). Kow, since eacii tooth ia to come into actioa
only when it comes into the line of centres, it is clear that
the tooth L must have been driven by K from the time when
their contact was in tiie line of centres, tmtil they have come
into the position shown in the figure, when the point of con-
tact of the anterior face of the next tooth O of the driving
wheel with the fiank* of the next tooth M of tlie driven
wheel has just passed into the Hne of centres ; and since the
tooth 0 ie now to take up the task of impelling the driven
wheel, and the tooth K to yield it, all that portion of the
lastrmentioned tooth which lies beyond the point B may evi-
dently be removed ; and if it he thns removed, then the tooth
K, passing out of contact, will manifestly, at that period of
the motion, yield all the driving strain to the tooth O, as it
is reqau'ed to do. In order to cut the pattern tooth of the
proper length, so as to satisfy
the proposed condition, we have
only then to take Aa (see the
accompanying figure) equal to
the pitch of the tooth, and to
biing the convex circumference
of the generating circle, so aa
to touch the convex circumfe-
rence of the arc MN in that
point a ; the point of intersec-
\ tioa e of this circle with the
Y face AE of the tooth will be
tlie last aotiiiff point of the tooth ; and if a circle be struct
from the centre of the pitch circle passing through that
point, all that portion of tlie tooth which lies beyond this cir-
cle may be cut off.-|-

The length of the tooth on the wheel intended to act with
this, may be determined in like manner.

210. In the preceding article we have supposed the same
generating circle to be used in striking the entire surfaces
of the teefti on both wheels. It is not however necessary to

* That portion of the edge of the tooth which ia vnthimt the pitch circle is
called its face, that inithin it ita Jlank.

t The point e thus determined will, in Bome cases, fall beyond the extremity
B of the tooth. In such cBsee it is therefore impossible to cut the tooth of
Buoh a length as to aatiafy the reciuired conditions, viz. that it shall drita only
after it has passed the line of centres. A full disouasion of these impossible
ca^ea wiU be found in Professor Willia'a work (Arts. 103-104.).

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248 TO DESCRIBE BPIOYOLOIDiL TEETH.

the correct working of tlie teetli, tliat tlie same circle should
thus be used in etriking tlie entire eurfaeea of ttwo teeth
which act together, but only that the generating circle of
every two portions of tlie two teeth whieh come into actual
contact should be the same. Thus the flo/rJc of the driving
tooth and the face of the driven tooth being in contact at

P in the accompanying figure,* this face of the one tooth
and flank of the other must be respectively an epicycloid
and a hypocycloid etinack with the same generating circle.
Again, the face of a drivuig tooth and the jlank of a di-iven
tooth being in contact at Q, these, too, must be stmck by
the same generating circle. But it is evidently imnee^aary
that the generating circle used in the second case should be
the same as that used in the first. Any generating circle
will satisfy the conditions in either case (Art. 202.), provided
it be the same for the epicycloid as for the hypocycloid
which is to act with it.

According to a general (almost a univei^al) custom among
mechanics, two different generating circles are thus used for
striking the teeth on two wheels which are to act together,
the diameter of the generating circle for striking the /aces
of the teeth on the one wheel being equal to the radius of
tlie pitch circle of the other wheel. Thus if we call the
wheels A and B, then the epieyeloidal faces of the teeth on
A, and the corresponding nypocycloidal flanks on B, are
generated by a circle whose diameter is equal to the radius
-of the pitch circle of B. The hypoeycloidal flanks of the
teeth on E thus become straight lines (Art. 203.), whose
directions are those of radii of that wheel. In like ii

is here supposed to drive tiie lowt

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TO DESCKTBE EPIOTCLOIDAI. TEETH. 249

the epieycloidai faces of the teeth on B, aiid the correspond-
ing hypocycloidal flanks of the teeth on A, are stnick by a
circle whose diameter is equal to the radius of tlie pitch cir.
cle of A ; so that the hypocycloidal flanlts of the teeth of A
become in like manner straight lines, whose directions are
those of radii of the wheel A, By this expedient of using
two different generating circles, the flanks of the teeth on
both wheels become straight lines, and the faces only are
cvirved. The teeth shown in the above figure are of this
form. The motive for giving this particular value to the
generating circle appears to be no other tlian that saving of
trouble wliich is offered by the subetitntion of a straight for
a cwved flank of the tooth. A more careful consideration
of the subject, however, shows that there is no real economy
of labour in this. In the flrat place, it renders necessary
the use of two different generating circles or templets for
striking the teeth of any given wheel or pinion, tlie curved
portions of the teeth of the wheel being strnck with a circle
whose diameter equals half the diameter of the pinion, and
the curved portions of the teeth of the pinion with a circle
wliose diameter equals half that of the wheel. Now, one
generating circle would have done for botii, had the work-
man been contented to make the flanks of his teeth of the
hypocycloidal foi-ms con'esponding to it. But there is yet a
greater practical inconvenience in this metliod. A wheel
and pinion thus constructed wiU only work with one another;
neither will work truly any third wheel or pinion of a differ-
ent number of teeth, although it have the same pitch. Thus
the wheels A and B liaving each a given number of teeth,
and being made to work with one another, will neither of
them work truly with C of a different number of teeth of
the same pitch. For that A may work truly with 0, the
face of its teeth must be struck with a generating circle,
whose diameter is half that of C : but tiiey are sti'uck with
a circle whose diameter ia half that of B ; the condition of
uniform action is not therefore satisfied. Now let us sup-
pose that &e epieycloidai faces, and the hypocycloidal flanks
of all the teetli A, B, and 0 had been stnick witb the same
generating circle, and that all three had been of the same
pitch, it IS clear that any one of them would then have
worked truly with any other, and that this would have been
equally true of any number of teeth of tlie same pitch.
Thus, then, the machinist may, by the use of the same gen-
erating circle, for all his pattern wheels of the same pitch, so
constnict tiiem, as that any one wheel of that pitch shall

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250 TO DE3CBIBE EPIOTCLOIDAL TEETH.

work with any other. This offera, under many cir cum stances
great advantages, especially in the very great reduction of
tlie number of pattems which he will he required to keep.
There are, moreover, many cases in whicli some aiTange-
ment similai' to this is indispensable to the true working of
the wheels, as when one wheel is required (which is often
the case) to work with two or three otners, of different num-
here of teeth, A for mstance to turn E and C ; by the ordi-
nary method of construction this combination would be
impracticable, so that the wheels should work truly. Any
generating circle common to a whole set of the same pitch,
satisiying the above condition, it may be asked whether
there is any other consideration determining the best dimen-
sions of this circle. There ie such a consideration arising
out of a limitation of the dimensions of the generating circle
of the hypocycloidal portion of the tooth to a diameter not
greater fiian half that of its base. As long as it remains
within these limits, the hypocycloidal generated by it is of
that concave form by which the flank of the tooth is made
f itself, and the base of the tooth to widen ; when
8 these limits, the flank of the tooth takes the con-
vex fonn, the base of the tooth is thus contracted, and its
strength diminished. Since then, the generating circlt
should not have a i^ameter greater than hdf that of any oi
the wheels of the set for which it is used, it will manifestly
be the greatest which will satisfy this condition when its
diameter is equal to half that of the least wheel of the set.
Now no pinion should have less than twelve or fourteen
teeth. Half the diameter of a wheel of the proposed pitch,
which has twelve or fourteen teeth, is then the tme diame-
ter or the generating circle of the set. The above sugges-
tions are due to Protessor Willis.*

* Profeeaor Willis hea suggested a new and very ingenious mettiod of
Btriking Ihe teelli of wheels by means of circular ores. A detailed deseriptioQ
of this method has been given by bim in the Transactions of the Institution
of CiTil Engineers, toI. ii., accooipanjed by tables, &c., which render its prac-
tjcal application esueeCingly simple and easy.

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TO DE3CEIBE IlfV'OLIITE TEEH^H.

311. To DEBCSIBB INTOLOTE TEETH.

Let AD and AG represent the pitch circles of
two wlieela intended to work together. Draw a
straight Hne FE through the point of contact A
of the pitch circles and inekned to the line of
centres CAB of th^e wheels at a certain angle
, FAO, the influence of the dimensions of which
I the action of the teeth will hereafter be ex-
plained, but which appears ueually to be taken
" not le^ than 80°.* Describe two circles ^EK
and yFL from the centi'es B and C, each touching the
straight line EF. These circles are to be taken as the hoses
from which the involute faces of the teeth ai'e to be stnick.
It is evident (by the similar triangles ACF and AEB) that
their radii CF and BE wiU be to one another as the radii
CA and BA of the pitch circles, so that the condition neces-
sary (Art. 201.) to the correct action of the teeth of tlie
wheels will be satisfied, provided their faces be involutes to

these two circles. Let AG and All in the above figure
represent arcs of the pitch circles of the wheels on an
enlarged scale, and .sE, fh, corresponding portions of the
circles eEK and /FL of the precedmg figure. Also let Aa
represent the pitch of one of the teeth of either wheel.
Tlu'ough the points A and a describe involutes ef and mn.\

* See Camus on the Teeth of Wheels, by Hawkins, p. 168.

I Mr. Haivkiiia recommends the following aa a convenient method of striking
involute teeth, in his edition of " Camus on the Teeth of Wheels," p. 166. Take
a thin hoard, or a plate of raetat, and reduce its edge MN so as aceuratoly to

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252 TO DESCKIBB IKYOLTrrK TEETH.

Let h be the point where tlie Une EF intersects the invohite
irm ; then if the teeth on the two wheels are to be nearly of
the same thickness at their bases, bisect the line A6 ui c ; or
if they are to be of different thicknesses, divide the line A&
in e in the same proportion*, and strike through the point o
an involute curve hg, similar to ^, but inclined m the oppo-
site direction. If the extremity ft' of tlie tooth be then cut
off BO that it may just clear the circumference of the circle
yL, the true form of the pattern involute tooth will be
obtained,'

There are two remarkable properties of involute teeth, by
tlie combination of ■which they are distinguished from teeth
of all other forms, and cmtf/n^s pa/rihus rendered gi'eatly pre-
ferable to all others. The lii'st of these is, that any two
wheels having teeth of the involute form, and of the same
pitch, f will work cori'ectly together, since the forais of tlie
teeth on any one such wheel are entirely independent of
those on the wheel which is destined to work with it (Art.
SOI.) Any two wheels with involute teeth so made to work
together will revolve precisely as they would by the aatnal
contact of two circles, whose radii may be found by divid-
ing the line joining their centj'ee in the proportion of the
radii of the generating circles of the involutes. This pro-
perty involute teeth possess, however, in common with the
epicycloidal teeth of different wheels, all of which are struck
■with the same generating circle (Art. 210.) The second no
less important property of involute teeth — a piopeit} which
distinguishes them from teeth of all other forms — is this
that they viork equaUy well, ho i vei faT the cerdies of the

de H h the cula a c
mi let a pe of th n
1 spr ng OE ha ing two
:l ng po nl9 upon it ea
1 t P and mb ch s of a
width equal w the thiclmegs of the plate be fixed upon edge bj means of
sL screw 0. Let the edge of tbe plate bo then made to coincide with the arc
eE in sucli a position that, when the epring is stretched, tlie point P in it may
coincide with the point from whioli the tootli is to be Btruek ; and the spring
beiiig itept contjmiallj' stretched, and wound or nnwouod irom the circle, the
iriTolute arc is thus to be deacribed by the point P upon the face of the board
from which the pattern is to be cat.

* This rule is pven bj Mr. Hawkias (p. 170,) ; it can only be an approKima-
tion, but may be BoffioientJj near to the truth for practical purposes. It is to
be obaerred that the teeth may have their bases in any other circles than
thosej/L and sE, from which the inrolutee are struck.

I The teeth being also of eqaal thiclinessea at their bases, the method ol
ensuring wMch condition has been espltuoed above.

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TSK TEETH OF A BACK AND PINIOH. 353

foheels are removed asunder' from one another ; so that the
action of the teeth of two wheels is not impaired when
their axes are displaced by that wearing of their brasses or
collars, wMcli soon results from a con-
tinued and a considerable strain. The
existence of this property will readily be
admitted, if we conceive AG and BH to
represent the generating circles or bases
of the teeth, and these to be placed witb
their centres 0, and 0, any distance
asunder, a band AB (p. 235., not«) passing
round both, and a point T in this band
generating a curve mw., m' n' on the plane
of each of the circles as they are made to
revolve under it. It has been shown that
tlieee curves irvn, and m! n' will represent the taces of two
teeth ■which will work truly with one another ; moreover,
that these curves are respectively involutes of the two
circles AG and BH, and are therefore wholly independent
in respect to their forms of the distances of the centres of
the circles from one another, depending only on the dimen-
sions of tlie circles. Since then the circles would di-ive at
any distance correctly by means of the band ; since, more-
over, at every such distance they would be driven by the
curves mn and m'n' precisely as by the band ; and smce
these cmTCs would in every such position be the same
curves, viz. involute of the two cirdee, ifc follows that the
same involute curves mn and m'n' would drive the circles
correctly at whatever distances their centra were placed ;
and, therefore, that involute teeth would drive these wheels
correctly at whatever distances the axes of tliose wheels
were placed.

The Teeth or A Eaok akd Pision.

312, To determine the pitch oirde of tlie pinion. Let H
represent the distance through which the rack is to be
moved by each tooth of the pinion, and let these teeth be
W in number ; then will the rack he moved iln-ough the
space N . H during one complete revolution of the wheel,
!N'ow the rack and pinion are to be driven by the action of
tlieir teetli, as they would by the contact of the circiira-

,y Google

J TEETH OF A HACK i

ference of tlie pitch circle of the
pinion with the plane face of the
rack, so that the space moved tlirough
by the rack dnrine one complete
revolution of the pinion must pre-
cisely equal the circumference of the
pitch circle of the piniou. If, tliere-
fore we call K the radius of the
pitch circle of the pinion, then

2*E=N . H ;

■.E=

-N . H.

213. To desoribe the teeth of the
pinion, those of the rack being
straight. The properties which have
been shown to belong to involute
teeth (Art. 201.) manifestly obtain,
however great may be the dhnensions of the pitch circle
of their wheels, or whatever disproportion
may exist between them. Of two wheels
OF and OE with involute teeth which
work together, let then the radius of tiie
pitch circle of one OF become infnUe, its
circumference will then become a straight
line repi'esented by the face of a rack.
Whilst the radius 0,0 of the pitch circle
OF thus becomes infinite, that C,B of the
circle from which its involute teeth are
struck (bearing a constant ratio to the fii'st)
will also become infinite, so that the invo-
lute m'n' will become a straight line* pei-pendicular to the
line AE given in position. The involute teeth on the
wlteel OF will thus become straight teeth ^ee^p'. 1.), hav-
ing their fac^ perpendiculai- to tlie line AB determined by
drawing through the point O a tangent to the circle AC,
from which the involute teeth of the pinion are sti'uck. If
the circle AC from which the involute teeth of the pinion
are struck coincide with its piteh cu-cle, the line AB becomes

" For it is evident that the extremity of a line of inSnile length unwinding
itaelf from the circumference of a circle of infinite diaraeWr will describe,
tiirougli a finite space, u str^ghl tine perpendicular to the circumference of
the circle. The Idea of giving an oblique position to the straight faces of the
teeth of a rack appears first to have occurred to Professor Willis.

, Google

parallel to the face of the rack, and the edges of the teeth
of the rack perjiendicular to its face {fig. 3.).

Now, the involnt« teeth of the one -wheel have remained
unaltered, and the truth of their action with teeth of the
other -wheel Jias not been influenced hy that change in the
dimensions of the pitch circle of the last, -which has con-
verted it into a rack, and its curved into straight teeth.
Thus, then, it follows, that straight teeth upon arack,_-work
truly -with involute teeth upon a pinion, indeed it is evi-

dent, that if from the point of contact P {fig. 3.) of such an
involute tooth of the pinion -with the straight tooth of a
rack we draw a straight line PQ parallel to the face a5 of
the rack, that straight line will be perpendicular to the
B-nrfaces of hoth the teeth at their point of contact P, and
that being perpendiculai' to tlie face of tJie involute tootli,
it will also touch the circle of which tliis tooth is the invo-
lute in the point A, at which tlie face db of the rack would
touch that circle if they revolved by mutual contact. Thus,
then, the condition shown in Art. 199. to be necessary and
eufficient to the correct action of the teeth, namely, mat a
line drawn from Hieir point of contact, at any time, to the
point of contact of theii' pitch circles, is satisfied in respect
to these teetli. Divide, then, the circumference of the
pitch circle, determined as above (Art. 212.), into N equal

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a56 THE TEGTH OF A EACK AND PINION,

parts, and describe (Art, 211.) a pattern involute tooth from
tile circumference ot' the pitch circle, limiting tiie length ot
the face of the tooth to a little more than the length BP of
the involute curve generated by unwinding a length AP of
the flexible line eqnal to the distance H through which the
rack is to be moved by each tootli of the pinion. The
straight teeth of the rack are to be cut of the same length,
and the circumference of the pitch circle and the face <w of
the rack placed apart from one another by a little more
than this length.

It is an objection to this last application of the involute
foi'm of tooth for a pinion workmg with a rack, that the
point P of tlie straight tooth of the rack upon which it acts
IS always the saine, being detennined by its intersection with
a line AP touching the pitch circle, and parallel to the face
of the rack. The objection does not apply to the preceding,
the case {fg 1 ) m "nhich the straight faces of each tooth of
the rack are inclined to one another. By the continual
action ii] on a single point of the tooth of the I'ack, it is
liable to an exeeetii^ e wearing away of its surface.

214 To describe the teeth of the pmion, the teeth of the rack
ieing cwved.

Tiiia may be done by giving to the face of the tooth of

the rack a CTcloidil toim md m il m^ thi Hee nt the tooth
of the pimon an epicycloid a'* ■nill be apparent if we eon
^ ceive the dnmetei of the ciicle whose

centre is C {see fig. p. 236.) to become
mfnite, the other two circles remain-
ing nnaltei'ed. Any finite portion of
the circumference of this infinite circle
will then become a straight line. Let
AE in the accompanying liguro repre^

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THE TEKPH OP A WHEEL WITH A lANTEKH, 257

sent such a portion, and let PQ and PR repreBent, aa
before, curves generated by a point P in the circle whose
centre is D, when aU three circles revolve by their mutual
contact at A. Then are PR and PQ the true forms of the
teeth which wonld drive the circles as they are driven by
tlieir mutual contact at A (Art. 202), Moreover, the curve
PQ is the same (Art, 199.) as would be generated by the
point P in the circumference of APH ; if that circle rolled
upon the circumference AQF, it is therefore an epieydoid;
and the curve PR is the same as would be generated by the
point P, if the circle APH rolled upon tlie circumference
or straight hne AE, it is therefore a eyaloid. Thus then it
appears, that after the teeth have passed the line of centres,
when the face of the tooth of the pinion is driving the flank
of the tooth of the rack, the former must have an epicy-
cloidal, and the latter a cycloidal form. In like manner, by
ti'ansferring the circle APH to the opposite side of AE, it
may be shown, that before the teeth have pa^ed the line of
centres when the flank of the tooth of the pinion is driving
the face of the tooth of'tJie wheel, the former must have a
hypocycloidal, and the latter a cycloidal form, the cycloid
having its curvature in opposite directions on the flank and
the face of the tooth. The generating circle will be of the
most convenient dimensions for the description of the teeth
when its diameter equals the radius of the pitch circle of
the pinion. The hypocycloidal flank of the tooth of the
pinion will then pass into a straight flank. The radius of
the pitch circle of the pinion is determined as in Art. 212.,.
and tile mf^tliod of describing its teeth is explained in
Art. 308.

£15. The teeth of j

In some descriptions of mill work the ordinary form of
the toothed wheel is replaced by a contrivance called a lan-
tern or tmndle, formed by two circular discs, which are con-
nected with one another by cylindrical columns called
staves, engaging, like the teeth of a pinion, ■with the teeth
of a wheel which the lantern is intended to drive. This
combination is shown in the following figure.

It is evident that the teeth on the wheel which works with
the lantern have their shape determined by the cylindrical
17

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THR 'ffiETH OF A WHEEL WITH A LANTEBR.

sliape of the staves. Tlieir forma may readily be found by
the metliod explained in Art. 300.

Having detennined upon the dimensions of the staves in
reference to the strain they are to be subjected to, and upon
the diameters of the pitch circles of the lantern and wheel,
and also upon the pitch of the teeth ; strike arcs AB and
^ AC of tliese circles, and set off upon thein
the pitches Aa and AJ from the point of
contact A of the pitch circles (if the teeth
krefifst to come into contact in the line
of centres, if not, set them off from the
points behind the line of centres where
the teeth are first to come into contact).
Describe a circle ae, having its eenti-e in
AB, paesi]ig through c^ and having its
^ ,1 to that of the stave, and (fivide each of the
iches Aa and Ah into the same number of equal parts
^say three). From tlie points of division A, a, /3 in the
pitch Affl, measure the sliorteet distance to the circle ae, and
■wi^ these shortest distances, respectively, describe from the
points of division 7, * of the pitch AS, circular area inter-
secting one another ; a cm-ve aS touchii^ all these circular
ancs will give the true face of the tooth (Art. 200.). The
■opposite face of the tooth must be struck from similar cen-
.tres, and the base of the tooth must be cut so far within the
pitch circle as to admit one half of the stave a£ -when that
stave passfts the line of centres.

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PRESSUEKS UPON WHEELS.

216. The kblaiios sbtWebs two PEBseuEfiS P, and P,

APPLIED TO TWO TOOTHED WHEELS IK THE STATE HOEDBE-
IHG UPON MOTION BY THE PREPONDEKANCE OF P,,

Let the influence of the weights of the wheels be in the
first place neglected. Let B and 0 represent the centres of
the pitch cirSes of the wheels, A their point of contact, P
the point of contact of the driving and driven teeth at any
period of the motion, EP the direction of the whole
resultant pressure upon the teeth at their point of contact,
which resultant pressure ia equal and opposite to the resist-
ance E of the follower to the driver, EM and OK perpen-
diculars from the centres of the axes of the wheels upon KP ;
and BD and CE upon the directions of P, and P,.

BD=a„ OE=a„ BM=m„ CN=ot,.

EA=r„ CA^r,.

Pn p5=radii of axes of wheels.

ip„ ip,=limiting angles of resistance between the axes of
the wheels and ■their bearings.

Then, since Pi and R applied to the wheel wliose centre ia

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260 BELATIOS OF THE DEIVIHG AUD WOEKIErQ

B are in the state tordering upon motion ty tlie preponder
ance of P„ and since a, and «», are the perpenaiciilars on
the directions of these pressures respectively, we have (eqna-
tion 158)

whore L, represents tiie length of the line DM joining the
feet of the perpendicnlara BM and ED.

Again, since E and P„ applied to the wheel whose centa-e
is C, are in the state bordering upon motion bj the yiddi/ng
of P, (Art. 16i.),

,.P-|^_(^)Mn.^^|E^l|^,-(^),i„.^,|E..(23T),

where L, represents the distance NE between the feet of the
perpendicularB CE and ON. Eliminating It between these
equations, we have

Kow let it be observed, that the line AP, drawn from the
point of contact A of the pitch circles to the point of contact
P of the teeth is perpendicular to their surfaces at that point
P, whatever may be the forms of the teeth, provided that
they act truly with one another (Art. 199.) ; moreover, that
when the point of contact P has passed tlie line of centres,
as shown in the figure, that point is in tl»e act of moving on
the driven surface V-pfrom the centre 0, or from P towards
f, so that the friction of that surface is exerted in the opposite
direction, or from p towards P ; whence it follows that the
r^ultant of this fnctiou, and the perpendicular resistance «P
of the driven tooth upon the driver, has its direction r2
within the angle oiPp and that it is inclined (Art. 141.) to the
perpendicular «P at an angle a^T equal to the limiting angle
of resistance. Kow this resistance is evidently equal and
opposite to the resultant pressm'e upon the surfaces of the
teeth in the state bordering upon motion ; whence it follows
that the angle EPA, is equal to the limiting angle of resist-
ance between the surfaces of contact of the teetii. Let this
angle be represented by p, and let AP=X, Also let the

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ioclinatioa PAG of AP to the line of centres BO be repre-
sented by B, Tlu-ougli A draw An perpendicular to EP, ami
sAt parallel to it. Then,

«i,r=BM=Bi + m=Pi:+A«,=BA sin. BA(+ AP eiu. APE-
Also BA(=BOP=PAC+APR=H?;

.-. m.^r, sin. (fl+(]))4-X gin. <f (^39);

«i,= C]?r=Cfi-sN=Cs-An=CA sin. OAs-AP sin. APE.
But As is parallel to PE, therefore CA«=B0E=9+<p;

.', OT,j=fj6in. (0+9)—'^ ein. ip (240.).

Substituting these values of m, and m, in the preceding
equation,

j',Bin.(S + ip)+Xsin,9+ I^J—ijsin. (p,

p.=

21 T. In tte preceding investigation tlie point of contact P

-P,...{2!U).

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263 EBlAnON OF THE DETVraG AND WOKKIHQ

of the teeth of the driving and di-iven wheels is suppceed to
have pa^ed the line of centres, or to be behind that line ;
let us now suppose it not to have passed the line of eentroa,
or to he before that line.

It is evident that in this case the point of contact P is ir
the act of moving upon the surface pPq of the driven tooth
towards the centre 0, or from P towards q, as in the other
case it h/rom the centre, or from P towards^. In this case,
therefore, the friction of the driven surface is exerted in the
dh'eetion qP ; whence it follows, that in this state bordering
upon motion the direction of the resistance R of the driven
upon the driving tooth must lie on the other side of the
normal APQ, being inclined to it at an angle APN equal to
the limiting angle of resistance. Thus the inclination of R
to the normal APQ is in both cases the same, but its position
in respect to that line is in the one case tlie reverse of its
position in the other case.*

The same consti-uction being made as before,
m,,=EM=Bi+M=B*+A7i=BA. sin. BAi+AP. sill. APO.

Also ]3Ai=E0R=BAP-AP0=^~P ;t
.'. m,j=r, sin. (3— >p) + ^ sin, ?,
m,=C]Sr=Cs-sN=:Cs-Aji=CA: sin. CAs-AP. sin. APO.
Eat As is parallel to PN,

.-. CAa=BOE=BAP-APO=S— p;
.'. TOj^r, sin. (fl— 9)— >^ein. ip.

Substituting these values of m, and m, in ecLuation (238),

1r,8in. (^— (p.)+>.Bin. ?+ I^^jsin. ip,
j^y— |-P,.(2i2).
^■jSin. (5— ^)—>. sin. 9—1^^1 sii

This expression differs from the preceding (equation 241)
only in the substitution of (d — 9) for (3+9) in the first terms
of the numerator and denominator.

* Hence it follows, that when the poliit of contact is in the act of crossing
the line of ceutrea, the direction of the resultant pressure E ie passing from
one side to the other of the perpendicular APQ ; and therefore that when the
point of contact is in tlie line of centres, the resultant preaanre is perpendicu-
lar to that line, and the angle BOR a right an^le ; a condition which oaimot
however be assumed to obtain approximately in respect to positions of any
point of contact exceedingly near to tlie line of centres.

t The angle 8 being here taken as before to represent the inclination BAP
of the line AP, joining the point of contact of the pitch circles with the point
of contact of the teeth, to the line of centres.

./Google

Dividing numerator and denominator of the fraction in
the Becond member of that equation by sin. (^ + 9), and
tJn-ovring out the factors r^ and t„ we have

= (-)

>-fiin. ip+ (!j— ij sin. 9,

",Bin.(fl4-t»)

/^^\

T'jsin. (^ + if>)

J

Now it is evident, that if in this fractional expression i—<p
be substituted for S +<p tlie numerator will be increased and
the denominator diminished, so that the value of^P, corre-
sponding to any given value of P,wiU be increased. Whence
it follows, that the resistance to the motion of the wheels by
tlie ftiction of the common surfaces of contact of their teeth
and of the bearings of tlieir axes is greater when the contact
of their teeth takes place lefore than when it takes place,
at an equal aiigular distance, lehind the line of centres — a
principle confirmed by the experience of all practical me-
chanists.

218. To DETEEMINE TnE RELATION OF THE STATE BOEnEEmO
UPON MOTIOS BETWEEN THE PEESSrEE P, APPLIED TO THE
DEIVINO WHEEL AND THE EESI3TAHCE Pj OPPOSED TO THE
MOTION OF TUE DEU'EN WHEEL, THE WEIGHTS OF THE
WHEELS BEING TAKEN INTO THE ACCOUNT.

Now let the influence of the weights "W, and "W, of the
two wheels be taicen into the account. The pressures applied
to each wheel being now three in nmnber instead of two, the
relations between P, and E, and P, and E are determined
by equation (163) instead of equation (158). Substituting
w, and "W, for P, in the two cases, we obtain, instead ot
equations (236) and (237), the following,

p,=r- •».+ R^

,„ M,W, . r---(2t3);

1 which equations M, and llj represent certain functions

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EELAl'ION OF THE DIUVDCG AND WOKKING

determined (Art. 166.) by tlie inclinations of the preseurea
P, and P, to the vei-tical.

Eliminating R betweon tlie above equations, neglecting
terms above tlie first dimensions in sin, 9, and sin. (p„ and
multiplying by afl,,

P,«,

- sin. 9, r — Pjtt, \

,M ,

, (344).

Substituting tbe Tallies of m, and m, from equations (239)
and (240), and neglecting tlie products of sin. 9, sin. 9, and
Bin. ip,, we obtain

V,a, \ r^sin. (S + 9)— Xsin. ip — -^ sin. <?, > —

V^aA r, sin. (3 + ip)+>- sill, 9 + -^ sm. 9, ^

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PEESSURES UPON WHEELS.

( L,'^.

263
. (245.)

,M,

Kow (Art, 166.) -~i=7n,cos. i^j + fii cos. !„, -where i^ repre-

eents the inclination "W,FPi of P, to the vertical, and i,, the
inchnfttion RrF of R to the vertical,*

Let the inclination "W.BD of the perpendicular upon P, to
the vertical be represented hy a„ that angle being so mea-
Bured that the pressure P, may tend to increase it ; let a, re-
present, in like manner, the inclination EGG of CE to the
vertical; and let /3 represent the inclination ABr of the
line of centres to the vertical,

.-. .„=:"W,FP,:="W.BD-BDF=a,-^,

i„=E7'F=:BOE-OE»-=d + 9-/3 ;

M

V — '=»!, sin. Kj + ffl, COS. (d+ip— /3),

Similarly —'=m,, cos. P,GH+ff, cos. P^W^.f Xow
P,GH=ECG-[-GEC=a,+|; and E*^"W,=*-Iij-E, and
PrF was before shown to be equal to (fl+9— /3 j
n. a, — «5 COS. (d+ip— ^)

K

Substituting tlie values of m, and m„ from eqiiations (239)
and (240),

— '=r, sin, (S-i-(p)sin. a, +>-sin. a, sin. ip4-
a, '

TjSin, (S+H)) sin. H^+Xein.ajBin.ip—
a, COS. (S+ip— /3)

M,

- See note, p. 113.

f It is W be observed that the direction of the a
BenW that of the cesietance opposed by the driten wi
driTiog wheel, so that tlie direction of the pressure
driyeii wheel is oppoate to that of the accow.

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266 BELATION OF THE DBITOTG AHD WOEKING

Let it be supposed that the distances DM and EN, repre-
sented by Li and Lj, are of finite diraensions, tlie direetiona
of neither of the pressures P, and P, approaching to coinci-
dence ■with the direction of E, — a supposition which has been
virtually made in deducing equation (163) from equation
(161), on the former of whicn equations, equations (243) de-
pend. And let it be observed that the terms involving sin, ^
in the above expressions (equations 246) will be of two di-
mensions in (;>„ ip, and ?, when substituted in equation (245),
and may therefore be neglected. Moreover, that in all eases
the direction of EP is so nearly perpendicalar to the line
of centres BC, that in those terms of equation (245), which
are multiplied by sin. ip, and sin. ip„ the angle i-\-<p, or BOB,

may be a8S8amed= ^ ; any error which that supposition in-
volves, exceedingly small in itself, being rendered exceed-
ingly less by that multiplication. Equations (246') will then
become

Substituting these values in the first factor of the seconr
member of equation (245), and representing that factor by
Nr,r„ we have

Nr^T-j^-^V^pj (r, sin. ttj -I- a^ sin. /3) sin, 9,—

and dividing by r^r^

N*=^-(ain. a,-l — ^ sin. /^)sin. ip,—
L, *"•

^^Hsin.a,-r^sin.^)sin.<p, . . . .(.247).

• If the direction of P, be that of a tangent at the point of contact A of
the wheels a ooae of frequent occurrence, the Talue of L, TaiuBlun^, that of N
would appear to become infinite in this eipreasion. Tbe difficultywill lioweTcr
be removed if we consider that when a^ beoomeB, as in this case, equal to r,,
and the point M is supposed to coincide with A, Li becomes a chord of the pitch

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IPKEBStrilES rPON WHEEI.S.

267

Substituting Kj',r, for the factor, -which it represents in
ecination (245), we have

?,«,{?•, sin. (fl+p)—>. sin. (p ip,Bin.ip,} — PAlAsiu-l^+fH

X sin. 9 + — sin. f,\ =.'Nr,r, sin. (^ +<f>) .
Solving this enuation in respect to P„

.(MS).

r, sin, (fl+9)

Xsiii.'p + -^Bm. p.

- ■ , MPs ■

Xsin. ?:H — ~6m.

"Whence, performing actual division hy the denominators of
the fractions in the second meniher of the equation, and
omitting terms of two dimensions in sin, <d„ sm. ip„ sin. <p
(ohservmg that !N" is already of one dimension in those vari-
ables), we have

circle, and is therefore represented bj 2f i eiii. iDBA, or 2ri sin. ^ (ni+i^) ; so

that ""• '""'"n""'^^ 8ii..ai-|-sin.3_2sin.i(.i.+^)cos.i(°.-/3)„
L, ~2r,ara.l(a,-|-/S) 2n ain. J (o.+i?)

If, tlierefore, we take the angle kj=:(3, so aa to give to Pi the direction of
s tangent at A, this expreasiou will assume the Talne, — cos. 0, or- ; so that

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i MODULC8 OF J

In this expression it is aBBUmed that tlie contact o
is behind the line of centves.

219. Tee modulus of a system of two toothed wheels.

Let % and «-, represent the numbers of teetb in thd
driving and driven wheels respectively, and let it be ob-
served that these nnmber are one to another as the radii of
the pitch circles of the wheels ; then, multiplying both sides

of equation (249) by «,~, we shall obtain

cosec.(S+?)i+Nr,.

Kow let A-), represent an exceedingly small increment of
the angle 4-) through which the driven wheel is supposed to
have revolved, after the point of contact P has pa^ed the
line of centres ; and let it oe obseiTed that the first member

of the above equation is equal to P,ffi,— -^. and that ~ a^-

represents the angle described by the driving wheel (Art.
204.), whikt the driven wheel describes the angle A-^-;

■whence it follows {Art, 50.) that P,ffi,[— A-l) represents the

work aU, done by the driving pressure P^ whilst this angle
^4- is described by the driven wheel,

^U, ^ (-,,(,/! 1 \ . , L^Pi ■ , I-iP, ■ .
• — ^r=^rMAl-\- ixi_4._ sra. flj-i — ^^ em. 9, -|- -^^-^ em. ?„!

cosec. {^ + 'p)[ +Nr,.

Let now A4. be conceived infinitely small, so that the first
member of the above equation may become the differential
co-efiicient of U„ in respect to 4'. Let the equation, then,
be integrated between the limits 0 and ^ ; P« L,, and L,,
and therefore N (equation 247) being conceived to remain

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or TWO TOOTHED WHEELS.

constant, whilst the angle -J/ is described ; we shall then
obtain the ec[uation

U.=P,«, /■ S 1 + IX (i + i^ sin. , + fe Bin. ,, + I* Bin. ,, j

cosee. (a + 9)l(;++:N".S (250),

where S is taken to represent the arc v^ described by the
pitch circle of the driven wheel, and therefore by that of the
driving wheel also, whilst the former revolves through the
angle +.

S20.. The moduli's op a system of two toothed wheels,

TIIE XUMBEE OF TliETH ON THE DEIVEN WHEEL BEIH& CON-
aiDEKAELE, AUD TIIE WEIGHTS OF THE WHEELS BEIHQ TAKEN
BTO ACCOUNT.

It is evident that the space traversed by the point of con-
tact of two teeth on the iace of either of tnem is, in this case,
small as compared with the radins of its pitch circle, and
that the direction of the resultant pressure K i^B^fig. p. 259.)
upon the teeth is veiy nearly perpendicular to the line of
centres EC, whatever may be the particular foiTus of the
teeth; provided only that they be of such forms as will
cause them to act truly with one another. In this case,
therefore, the angle EOR represented by ^+<p is very nearly

ec[nal to -, and cosec. (fl+ij))=l.

Since, moreover, RP is very nearly perpendicular to the
line of centres at A, and that the point of contact P of the
teeth deviates hut little from that line, it is evident that the
line AP represented by X differe but little from an arc of
the pitch circle of the driven wheel, and that it diffei-e the
less as the supposition made at the head of this article more
nearly obtains. Let us suppose ■\' to represent the angle
subtended by this arc at the centre 0 of the pitch circle of
the driven wheel, then will the arc itself be represented by
r^-^i, and therefore 'k^=T^-\' very nearly. Substituting this
value of >- in equation (350), observing that cosec. (fl + ip)=l,

and that — = — (eq^nation 327), and integrating,

U,= Sl + i4'(l+'^Uin.<p+^Bin.9.+^'siu.a',f
\ nj a,ri a,r, '

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270 INTOLrTE TTiETH.

P,a,+ + Xr,4. (251).

But the driven or working preeeure P, teing eupposed tc
remain constant, wliilst any two given teeth are in action,
Fjfflj-i' represents the work U, yieloed by that pressure whilst
those teeth are in contact : also r,-!' represents the space S,
descnbed by the circumference of the pitch circle of either
wheel whilst this angle ia desevibed. Now let ■^^ be con-
ceived to represent the angle subtended by the pitch of one
of the teeth of the driven wheel, these teeth being supposed

to act only leMnd the line of centres, then 4'= — , n, repre-
senting the number of teeth on the driven wheel, and ^4-

1 nj n,\ 11,1 In, nj'

.■,U.= il+.(i+i)8m.»+il''-!Bm.?.+ i*sm.».l

U,+ N.S (253),

which relation between the work done at the moving and
working points, whilst any two given teeth are in contact, ia
evidently also the relation between the work similarly done,
whilst am/ gwen number of teeth are in contact. It is there-
fore the MODDLus of any system of two toothed wheels, the
numbers of whose teeth are considerable,

221. Tm: MODm.us of a st

LUTE TEETH OF ANY

The locus of the points of contact of the teeth has been
, — -.^ shown (Art. 201.) to be in this case

~ ~" a straight line I>E, which passes

through the point of contact A of
the pitch circles, and touches the
circles (EP and DG) from which the
involutes are struck. Let P repre-
\ \ sent any position of this point of
i contact, then is AP measured along
tlie given line DE the distance re-

'"x""---- -■'',•'' presented by X in Art. 216., and the

~'"-'^'"'' angle CAD, which is in this case

constant, is that represented by S. Since, moreover, the
point of contact of the teeth moves precisely as a point P
upon a flexible cord DE, unwinding from the circle EF and
winding upon DG, would (see note, p. 235.), it is evident

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INVOLUTE TEETH. 271

that the diatance AP, being that which such a point, would
traveree whilst the pitch circlo AH revolved through a cer-
tain angle ^^, measured from tlie Kne of centres is precisely
equal to the length of string which would -wind upon DG
whilst this angle is described by it; or to the are of that
circle which subtends the angle +- Il^i therefore, we repre-
sent tlie angle ACD by v, so that CD=CA cos. ACD=r,
cos. 1, then >-=:r,4' cos. ^. Substitating this value for X in
equation (249), and observing that 6-\-(p = - — ii+(p = - —

(1—9), and tliat — = — , we have

1

sec. (>)— ip)yP,-l~:

Nn

.(253);

from which equation we obtain by the same steps as in
Art. 219, obsei-ving that n is constant,

U,= }l-1- Wi — 1- — Icos. V sin, ip-H^^sin. Ti, + -^sin, ?,}

:.(»-?) lu,+NS (354),

which is the modulus of a system of two wheels having any
given numbers of involute teeth.

222. The involttte tooth of least kesistahoe.

It is evident that the value of U, in equation {254), or of
^ the worh which must be done

.-''.'-■.;'^, .upon the drivingwheel to cause

a given amount U, to be yielded
by the driven wheel is dependent
for its amount upon the value of
the co-efficient of 11, in the
second member of that equation ;
and that this co-ef&cient, again, is
dependent for its value (other
things being the same) upon the
value of 1 representing the angle
ACD, or its equal the angle DAI,

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272 THE INVOLUTE TOOTH OF LEAST

which the tangent DE to the chcles from whicli the invo-
lutes are etrack makes with a perpendicular AI to the Hne
of centres. Moreover, that the co-efRcieat N" not inTolving
this factor i (equation 347), the variation of the value ot
U„ so far as this angle is concerned, is wholly involved in
the cori'esponding variation of the co-efficieut of TJ, and
imes a minimum with it ; so that tlie value of *i which
8 to the function of i represented hy tliis co-efficient, its
1 value, is the value of it which satisfies the condi-
tion of the greatest eoonoTivy of power, and determines that
inchnation DAI of the tangent I)E to the perpendicular to
tlie line of centres, and tliose values, therefore, of tlie radii
CD and BE of the circles whence the involutes are struck,
which correspond to the tooth of least resistance.

To detei-mine the value of i which corresponds to a mini-
mum value of this co-efficient, let the latter he represented
by u ; then, for the required value of i,

-7-=0, and j-;>0.
Let*l-H — h=A,-i^siu. 9,+ -^sm. f„=B:

;.«— 1 + (A cos. 1 sin. ip-l-B) sec. (11—9);
.'.w=l-|-B sec. (n— q>) + A sin. 9 cos. ■i\ see. (^—9);

.■.-T-=B sec. ('1— 9)tan. (ii— ffl)— Aaln. ipjsiu.i sec. (1— ^)—

cos, 5] tan, (1— -ip) sec. {1— 9)} ;

.•.-T-=B sec. Xi— 9) sin. (ii—o))— ■

A sin. 9 sec. '('I— 9)^^. ^ cos. ('i— 5)— cos. 1 sin. (1— p)} ;

.•.-T-=sec. '(11— (p)jB sin. (ii— ip)~-A sin. '9} (255).

1 for any value of

■n, one of the factors which compose the second memher of
tlie above equation must vanish for that value of 1 ; but
tliis can never be the case in respect to the first factor, for
tlie least value of the square of the secant of an arc is the
square of the radius. If, therefore, the function u admit of

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THE INVOLUTE TOOTH OF LEAST EESISTAKCE.

2t3

a minimura value, tlie second factor of the above equation
vanishes when it attains that value ; and the corresponding
value of V is detennined by the equation,

B sin. (i— (p)-A sin. \=0 (256).

or by sin. (^i— ?)=^sin. V or by i]=9+sin. ( gSin. V);
or substituting the values of A and B,

.(25t).

Now the function u admits of a minimum to which this
value of -1 corresponds, provided that when substituted in

-ri this value of i gives to that second differential co-effi-
cient of u in respect to i a.posiiive value.
Differentiating equation (353), we have

-^"5=2 sec. '('1— *■) tan. (i— (p){B sin. (ii— ip)-

A sin. Vi +B sec. '{n—if) cos. ('i— 9)

But the proposed value of »i (equation 256) has been'
shown to be that which, being substituted in tiie factor {B!
ein. (i]— ?)— A sin. \\, will cause it to vanish, and therefore^

with it, the whole of the first term of the value of t~s : it

an
corresponds, therefore, to a minimum, if it gives to tlie-
second term B sec. '{n—f) cos. (1—9) a positive value ;, or,
since sec. \ii—<p) is essentiallv positive, and B does not
involve 1, if it gives to cos. (»i— f) a positive value, or if

(p<5 0rifBin. ( ^ ein. V <^, or if wsin. '?<! ; oril

A ein. '(p<B; or if

\B°

'B°

1 sin. '9< -^-^ain.^i + -^^6in. ?,.,.. (258).

Tills condition being satisiied, the value of *j, determinedi

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274 THE BEST DIViaiOK OF THE ANGLS OP CONTACT,

hj equation (257), correBponds to a minimuin, and deter-
mines the ISV0LT3TB TOOTH OF LEAST RESISTANCE.*

223. To DETERMESB IX WHAT PKOFOKTIOK THE AHGLB OF
CONTACT Olf EACH TOOTH BHOTILD BE DIVIDED BY THE LINE
OF CENTERS ; OK THROUGH HOW MUCH OF ITS TITCH EACH
TOOTH SHOULD DKIVB BEFORE ASD BEHIND THE LINE OF
CENTEES, THAT THE WORK F.XPENDBD UPON FKICIION MAT
BE THE LEAST POSSIBLE.

Let the proportion in which the angle of contact of each
tooth 18 divided by the hne of centres be represented by x,

BO that !B^ may represent the angular distance from the line

of centres of a line drawn from the centre of the driven
wheel to the point of contact of the teeth when they first

3*
come into action before the line of centres, and (1— a?) —

the corresponding angular distance behind the line of centres
when they pass out of contact ; and let it be observed that,
on this supposition, if U, represent as before the work
yielded by the driven wheel during the contact of any two
teeth, icU, will represent the portion of that work done
before, and (l—is)U, that done behind, the line of centres.
Then proceeding in respect to equation (253) by the same
method as was used in deducing from that equation the
modulus (Equation 254), hut integrating first between the

limits 0 and x — , in order to determine the work u, done by

the driving pressure before the point of contact passes the

line of centres, and then between the limits 0 and (1— i") — ■

to determine the work u, done after the point of contact lias
passed the line of centres ; observing moreover, that in the
former case —? is to be substituted in see. (»?— 9) for 9 (Art.
217,), we have

* Tt may easily be shown by eliminating ti between equations (254) and
(256) that the modulus corresponding to this condition of the greatest economy
,of power, where involute teeth are used, is represented by the formula

Ui- j l-f-JA sin. 2(S+(B"~A' sin. V) [Ws + KS.

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r DIVieiOH" OF THE ANGLE OF CONTACT. 275

--^sin. %\ sec. (1+9} f icTJi+Ns, ;
Or assuming

/■I 1\ . ,I',P. ■ L,Pi. 7.

ffl — I — Icos.'! sm. (p=:c(, and-^sm. <p, + — sm. 9j=o

«, representing tlie space described by the pitch circle of
either wheel before the line of centres is passed ; Bimilarly,

«,= j 1+ \a{l-x) + l>\ sec. {■•)-'?) I (l~a;)XJ,+Ns,.

Adding these equations together, and representing by TT, the
whole work u, + u, done by the driving pressure during the
contact of tlie teeth, and by S the whole space described by
the circumference of either pitch circle, we have

lj,= I l+iax' +dx)se<!.{f]+f)-i-

Sa(l-«)'+5(l-ic)}sec,(^-?)|u.+NS . . . (359)

by which equation is determined the modulus of two wheels
driven by involute teeth, when the contact takes place partly
before and pai'tly behind the line of C'

Let the portion of the work U„ which is expended upon
the friction, of tiie teeth be represented by u. Then

w^ \ ((iie''-|-53i)sec.('i+(p) +

|a(l-;cy+J{l~«)fsec. {n-v) ] U,+NS.

Now the value of x, which gives to this function its mini-
mum, and which therefore determines that division of the
driving arc which corresponds to the greatest economy of
power, is evidently the value which satisfies the condition

dat ax

But differentiating and reducing

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276 THE BEST DIVISIOK OF THE ANGLE (

5jsee. (^ + ?)— sec. {>!— ip)j — 2(t eec. (i— tp) [ 0,;

--T-j=2ajeec. (i]+ip) + sec. {i— ip)|TJ, :

Whence it appears that tlie second condition is always aatie-
fled, and that the first condition is satisfied by that value of
a?, which is determined by the equation

2(Ke{9ec, {^+ip) + sec. (^— (p)} +Sjscc. (1 + 9)— eec. (1—9)} —
Srasec. (^— ?')=0;
Whence we obtain by transposition and reduction

iB=-| 1— 11+- jtan. »]tan. p [ .

So that the condition of the greatest economy of power ia
satisfied in respect to involute teeth, when the teeth fii-st
come into contact before the line of centres at a point whose
angular distance from it is less than one half the angle sub-
tended by the pitch by that fractional part of the last-men-
tioned angle, which is represented by the foiinula-^ln — ]

tan. 7} tan. a, or substituting for J and a their values by tlio
formula

— sin. <p-\ — ~i

1 + '^''°' -^ — T\^'^~ \ '^^'^- 1 1^"- ^ ■ • • (^^^)-

*(-+-p

That division of the angle of contact of any two teetli by
the line of centres, which is consistent with the gi-eatest
economy of power, is always, therefore, an unequal division,
the less portion being that which lies before tlie line of cen-
tres ; and its fractional defect from one half the angle of con-
tact, as also the fractional excess of the gi-eater portion above
one Jialf that angle, is in eveiy case represented by the above
formula, and is therefore dependent upon the dimensions of
the wiieels, the forms and numbere of the teetli, and the cir-
enmstanees under which the driving and working pressures
are applied to tiiem.*

" The diviaion of the arc of contact which correaponda to the greatest eco-
nomy of power ill cpicycloidal teeth, may be aetermmeil by precisely the s«nie

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THE MODUma OF A BYSTKM OF TWO WHEELS.

224. The modulus of a system of two wheels driven j

The locus of the point of contact P of any two such teeth
is evidently the generating circle APH of
the epicycloidal tace of one of the teetli, and
the hypocycloidal flanit of tlie other (Art.
202.) ; for it has been shown (Art. 199.),
tliat if the pitch circles of the wheel and the
generating circle APH of the teetli be con-
ceived to revolve about fixed centres B, C,
D by their mutual contact at A, then will a
point P in the circumference of the last-men-
tioned circle move at the same time upon
the surfaces of both the teeth wliich ai'e in
contact, and therefore always coincide with their point of
contact, so that the distance AP of the point of contact P of
the teeth from A, which distance is represented in equation
(250) by \ is in this case the chord of the arc AP, which
the generating circle, if it revolved by its contact with
the pitch circles, would have described, whilst each of the
pitch circles revolved through a certain angle measured
from the line of centi-es. Let the angle which the driven
wheel {whose centre is 0) describes between the peiiod
when the point of contact P of the teeth passes the line of
centres, and that when it reaches the position shown in the
figure be represented as before by -Jj, the arc of the pitch
eu'cle of that wheel which passes over the point A during tltat
period will then be represented by r,-^. Now the generating
circle APH having revolved in contact with this pitch circle,
aneqnalarcof that circle will have passed over the point A;
whence it follows that the arc AP=r,4' ; and that if the radius
of the generating circle be represented by r, then the angle

ADP subtended by the arc AP is represented by -^l; or

by 2^, if 2e be taken to represent the ratio — of the radius

of the pitch circle of the driven wheel to the radius of the
g&nerating circle. Now the chord AP=3AD sin. -J ADP;

therefore >^=2y sin. 64-=— sin. frj-- Substituting this value

of >^ in equation (250) ; observing, moreover, that the angle

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278 THE MODTILirS OF A SYSTEM OF TWO 'WHEELS

PAD represented by ^ in that equation is equal to^ — -J

ADP, or to 5— «+> and that the whole angle 4- through
which the diiTen wheel is made to revolve by the contact of
each of its teoth is represented hy— , we have

XJ,=F,a,r \ 1+ f^i—-{.—\ sm.? sin. «4' + -!^ sin. <p,+

^ sio.^Jsec. (e^^-?) I di^+NS ;

or, assuming L, and L, to remain constant duiing the cuu-
tact of any two teeth representing the constant 1 + -—sin.ip, +

Ui=Pa I A /'sec. (e4'— f)i^4' + -|l+— j sin. (p/*Bin. ^ sec.

Kow the general integral, / sec. {e^—tf^d-l; or
- / sec. {e^—^)d{&\'—<l>) being represented* by the function

i_ /'COB. {e'p-4)d{etp-^ 1 /•cos.(<'''-0)d(#-^) 1

ej l-Mn.'{e'/>-« "as J l-|-^in. W-W '''

I- [ei^-1*) ^ =--1

I 1-i-ain. (^iJ--») I L
1 iTain. (.v--- ,1) f

./Google

1 EPiCYOLOIDAL TEETH. 279

- log. tan. I j+i(M'—'P) \ > ite definite integral between the
limits 0 and — has for its expression,

ilog. ^* "' "' '-L

'"'• (j-l)

Also/sec.{e4'— 'p)sin.«4"^=/eec.(e4'— ?)sin. {(^4— 9')+ii>f(H

= /sec. (e-l-— ®) Ssin. {e-^—f) cos. p + cos. (fi^-— ip) sin.ipjfi'}'

= / 5eoB. 9tan. (^4'— ?)+siii. 9|(^

•in
=-co9. ?>/ tan. (c4-— 9) (^ (e4'~'P)+~" sin. tp-

Now the general integral / tan. (^4'— 9)'^(^4'— *P) bas for
its expression— log. ^cos. (s-l'— ?).* Taking its definite inte-
gral between tlie limits 0 and — , we have, therefore,

/"* 1 '^°^'\~^ I 2^

see. (^4^— 'p)ain. e4'(?4'= — coa.iplog, ' ' +— ^sin-ip.

^ ^ COS. "P "a

/— [Zcoa. (jV— ^)__

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THE MODTJLUB OF A SYSTEM OS 1

Substituting these expreesioiiB in the modulus, representing
J— I ^y ?'i ^^ observing tliafc if U, represent the work
yielded by the driTen wheel during the action of each tooth,
then Pa-— =U„ so tliat P,a,=^'., we have

-e^^i

I 1 U.+HS . . . (361).

/2eir \

COS. ip '"a

,-j. , cos.( — — (p) i 1 , . 2e\ ) 2e«

.Kow log. .\ »' _/ = log. { 1+tan. — tan.ip > cos, — ~

' C0S.9 't '^^ ' '^^

iog.^cos.-^ — +'''S-s )l + tan.- — - tan. O) > = log.^ cos. +

tan. tan.ip— ^ tan,' , tan,°ij> + &e. Substituting this

expression in the preceding equation, and neglecting terms
above the first dimension in tan. 9 and sin. 9,

.^lu,+NS (262).

225. If the radius r of the generating circle be eyual to
one half the radius r, of the pitch circle of the driven wheel,
according to the method generally adopted by mechanics

(Art. 203.), then e=i^ = i~=l.

In this case, therefore— that is, where the flanks of the
driven wheel are straight (Art. 210.) — the modulus becomes

U,+NS .

— ^ 1 1 + — 1 sin. 2xi log.

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HAVING EPICTCLOIDAL TEETH.

■ Substituting (in equation 263.) for v' its value _

2 tan.| +itan.'(J-|)+|tan.=|+&c.

If, therefore, we assume the teeth in the driven wheel tc
be so numerous, or n^ to be so gi'eat a number, that the third

power and all higher powei-s of tan, ( 1) may be ne-

glected as compared with its first power, and if we

powera of tan. - above the second.

log.B^ — / ' .— ■■ =2 \ tan. I nl+tan.^ \

= ' tan in \ii, 9.1 9

which expression becomes — if we suppose the two arcs
which enter into it to be so small ae to equal their respee-

* Por aeaujne log.E cos. a=BiST'+Bi3T*-f-!'sa:°+ . . . . ; then differontiaticg,

-ton. j:=2a,a!-i-4«,a'+6a,a!'+ ;

but (Miller, Dif. Gal. p. 96.)— tan. x^—x—ix'—^—rx'—. . . . ; equating,
thereforo, the co-efSoients of these identical aeries, we haye

, Google

3 MODULUS OF THE BACK j

Substituting these vahiee in ecLuation (362), and perform-
ing actual multiplication by the factor -n^, we have

U,= { A+i*(^ +^) sin. 2? I TJ,+NS ;

' for A its value ; and assuming ^ sin. 2?=
Bin. (p, since ? is exceedingly small,

U.= |(l + ^sin. ,.+^sin.9.) +
<(^+^)sin.9 }l\+NS (264),

■which is the modulus of a wheel .and pinion having epicy-
cloidal teeth, tlie number of teeth n^ ni the driven wheel
being considerable (see equation 353).

It 18 evident that the value of XT, in the modulus (equa-
tion 361), admits of a nmmrmm in respect to the value of e;
there is, therefore, a given relation of the radius of the
generating circle of the driving, to that of the driven wheel,,
which relation being obsei-ved in striking the epicycloida?
faces and the hypoeycloidal flanks of the teeth of two wheelu
destined to work with one another, those wheels will work
with a greater economy of power thtm they would under any
other epicyeloidal forms of their teeth, lliis value of e may-
be determmed by assuming the differential co-efficient of the
co-efficient of U, in equation (261) equal to zero, and solving
the resulting transcendental equation by the method of

327. The modtjlhs of the eack isn rmioN.

If the radius r^ of the pitch circle of the driven wheel be
supposed infinite (Art. 218.), that wheel becomes a rack, and
the radiuB r^ of the driving wheel remaining of iinite dimen-
sions, the two constitute a rack and pinion. To determine
the modulus of the rack and pinion m the case of teeth of
any form, the number upon the pinion being great, or in
the case of involute teeth and epicyeloidal teeth of any
number and dimensions, we have only to give to r, an
infinite value iu the moduli already determined in respect

./Google

TIIS MODfLTIS OF THE EACK AND PINIOtT. 283

to these several conditions. But it la to be observed in
respect to epicyeloidai teeth, that «., becomes iuiinite with

r„ whilst the ratio— remains finite, and retains its equality

to the ratio — (equation 327), so that— = ^ — =^ — ^ =^ ;

if we represent the ratio — by 2e,. Making n, and r, infinite
in each of tlie equations (252), (254), and (261), and sub-
stituting — for — in equation (262) ; we liave

1. For the modulus of the rack and pinion when the teeth
are veiy small, whatever may be their forms, provided that
they work tnily.

\J ~ \ 1 + ^sin. 9, + -sin. 9 i U,+NS .■ (265).

3. IFor the modulus of a rack and pinion, with involute
teeth of any dimensions (se&fy. 1. p. 355),

U,= j 1+ I — eos.)7sin.ip + — ^'ain. 0,1 sec. (jj— 0) [111+

N8 . . (266).

3. Por the modulus of the rack and pinion, with cycloidal
and epicycloidal teeth respectively (equation 261),

1 +^'sin.0jlog.£

2e, * COS. 9 > '

In each of which cases the value of N is determined by
making r^ infinite in equation (34T).

11 of t!ie rack upon it3 guide

, Google

CONICAL WHEELS.

Conical ok Bevil Wheels.

228. These wheels are used to couimuiiieate a motion of
rotation to any given axis from another, inclined to the first
at any an^le.

Let AF be an axis to which a motion of rotation is to be
communicated from another axis AE
inclined to the fii-at at any angle EAF,
by means of bevil wheels.

Divide the angle EAF by the straight
line AD, so that DO and DW, perpen-
diculars from any point D in AD upon
AE and AF respectively, may be to
one another as the numbers of teeth
which it is required to place upon the
two wheels,*

,.^ _. 8 to be generated by the revolution of the

line AD about AE, and another by the revolution of the
line AD about AF. Then if these cones were made to
revolve in contact about the fixed axes AE and AF, their
surfaces wonld roll upon one another along their whole line
of contact DA, so that no part of the surmee of one would
elide upon that of the otJier, and thus the whole surface
of the one cone, which passes in a given time over the line
of contact AD, be equal to the whole surface of the other,
which passes over that line in the same time. For it is
evident that if n, times the circumference of the circle DP
be equal to «., times that of the circle DI and these circles
be conceived to revolve in contact carrying the cones with
them, whilst the cone DAP makes j*, revolutions, the cone

of t)ie angle EAF may be made as fpllowa : — Draw ST and
' 'J in the atraight lines AE imd AF at right angles
to those lines raspeetiTely, and haying their
lengths in th« ratio of the nmnberB of teeth
whidiitisreqnired to place upon the two wheels;
and through the extremities T and W of these
hnea draw TD and WD parallel to AE and AP
respeelivelj, and meetmg in D. A straight line
drawn from A through D will then make the
required division of the angle ; for if DO and
DN be drawn perpendicular to AE and AF, they
will evidently be equal to UW and ST, and there-
fore in the required proportion of the nuuibera
of the teeth ; moreover, any other two lines
drawn perpendicidur to AE and AF from any
■fpstlj be in the same proportion as PO and PS.

./Google

DAI will make n^ revolutions ; so that whilst any other
circle GH of tlie one cone makes «-, revolutions, the corre-
sponding circle HK of the other cone will make n, revolu-
tions : but «., times tlie circumference of the circle GH
is equal to n, times that of the circle HK, for tlje diametei-s
of tiiese circles, and therefore their circumferences, are to
one another (by similar triangles) in the same proportion as
the diameters and the circumferences of the circles DP aud
DI. Since, then, wliilst the cones make n^ and n, revolutions
respectively, the circles HG and HK are canned tlirough n^
and n, revolutions respectively, and that n, times the circum-
ference of HG is equal to n^ times that of HK, therefore
the circles HG and HK roll in contact through the whole of
that space, nowhere sliding upon one' another. And the
same is true of any other corresponding circles on the cones ;
whence it follows that their whole snrfaces are made to roll
upon one another by their mutual contact, no two parts
being made to slide upon one another by the rolling of the
rest.

The rotation of the one axis might therefore be commmii-
cated to the other by the rolling of two such cones in con-
tact, the smface of tlie one cone carrying with it the surface
of the other, along the line of contact AD, by reason of the
mutual friction of their surfaces, supposing that tliey could
be so pressed upon one another aa to produce a friction equal
to the pressure under which the motion is communicated, or
the work transferred. In such a case, the angular velocities
of the two axes would evidently be to one another (equation
227) invereely, as the circumferences of any two correspond-
ing circles DP and DI upon the cones, or inversely as their
radii ND and OD, that is (by construction) inversely as the
numbei's and teeth which it is supposed to cut upon the
wheels.

"When, however, any considerable pressure accompanies
the motion to he communicated, the friction of two such
cones becomes insuflicient, and it becomes nece^ary to
transfer it by the intervention of bevil teeth. It is the cha-
racteristic property of these teeth that they cause the motion
to be ti'ansterred by their successive contact, precisely as it
would by the continued contact of the stirfaces of the

./Google

229. To describe the teeth of bevil wheels*

From D let FDE he wrawii at riglit angles to AD, inter-
secting the axes AE and AY of the two cones in E and F ;
suppose conical surfaces to be generated by tlie revolution
of the lines DE and DF about AE and AF respectively ;

\4

and let these conical surfaces be truncated by planes LM
and XY respectively perpendicular to their axes AE and
AF, leaving the distances DL and DY about equal to the
d&pths which it is proposed to assign to the teeth. Let now
the conical surface LDPM be conceived to be developed
upon a plane perpendicular to AD, and passing through the
point D, and let the conical surface XIDY be in like
manner developed, and upon the same plane. When thus
developed, these conical surfaces will have be-
come the plane surfaces of two segmental annuli
MPpm and IXtraf, whose centres are in the
points E and F of the axes AE and AF, and
f which touch one another in the point D of the
line of contact AD of the cones.

Let now plane or spur teeth be struck upon
i the circles rp and li, such as would cause them

* The method here given appears first to have been published by Mr. Tred-
golii in his edition of Buchanan's Essay on Mllvm-Ic, 1823, p. 108.

f The lines MP and pm in the dGrelopment, coincided upon the cone, as also
the lines IX and ix; the other letters upon the devulopnient in the above

./Google

CONICAL WHEELS. 287

to drive one another as they would be driven ty their
mutual contact ; that is, let these circles F» and I* be taken
ae the pitch circles of sach teeth, and let the teeth ho
describea, hy any of the methods before explained, so that
they may drive one another correctly. Let, moreover, their
pitches be such, that there may be placed as many snch
teeth on the circumference Pp as there are to be teeth
upon the bevil wheel HP, and as many on J.i as upon the
■wlieel HI.

Having struclc upon a flexible surface as many of the first
teeth as are necessary to constitute a pattern, apply it to
the conical surface DLMP, and trace off the teeth from it
upon tl)at suiface, and proceed in the same manner with tlie
surface DIXY.

Take DH equal to the proposed lengths of the teeth, draw
ef through H pei-pendicnlar to AD, and, terminate the wheels
at their lesser extremities by concave surfaces HGrmi and
HKot described in the same way ae tho convex surfaces
which form their greater exti-emities. Proceed, moreover,
in the construction of pattern teeth precisely in the same
■way in respect to those surface as the other ; and trace out
the teeth from Uiese patterns on the lesser extremities as on
the greater, taking care that any two similar points in the
teeth traced u]dou the greater and lesser extremities shall lie
in the same straight line passing through A, The pattern
teeth thus traced upon the two extremities of tlie wheels are
the exti-eme bomidaries or edges of the teeth to be placed
upon them, and are a sufficient guide to the workrnan in
cutting them.

230. To j}rove that teeth thus constructed wiU work truly
with one anoth^.

It is evident that if two exceedingly thin wheels had been
taken in a plane perpendicular to AD {Jig. p. 286.) passing

figure represent points wbioh are identical with those shown by the Eame let-
Ceis in the preceding Hgure. In that ligure tlie conical surfaces are ehonn
developed, not in a pkne perpendionlar to AD, but in the plane which contains
that line and the lines A.£ and AF, and which is perpendicular to the last-oieu-
tioned pluiie. It is evidentty unnecessary, in the construction of the patt«m
leetli, actually to derelope the oonical estremities of the wheek aa above
described ; we have only to determine the lengths of the radii DE and DF by
construction, and with them to describe two arcs, Pp, li, for the pitch eirclea
of the l«eth, and lo set oft' the pitches upon them of the same lengths aa the
pitches npor, the circles DP and DI, which last are determined by the numbeis
of teeth required to be cut upon the wheels cespoctivcly.

, Google

THE MODULUB OF A 6YBTEM

through the point D, and having their centres in E and F ;
and if teeth had heen cut upon these wheels according tc
the pattei'n above described, then would tliese wheels havo
worked truly with one another, and the ratio of their angu-
lar velocities have been inversely tkat of ED to ED, or (by
siniilar triangles) inversely that of ND to OD : which ie the
ratio required to bo given to the angular velocities of tlie
bevil wheels.

Now it is evident that that portion of each of the conical
sai-faces DPML and DIST which is at any instant passing
through the line LY is at that instant revolving in the plane
pei^jendicular to AD which passes throngh the point D, the
one surface revolving in that plane about the centre E, and
the other aboat the centre F ; those portions of the teetli of
the bevil wheels which lie in these two conical suri'aees will
therefore drive one another truly, at the instant when they
are passing through the Une LY, if they be cut of the forma
which they must have had to drive one another truly (and
with the required ratio of their angular velocities) had they
acted entirely in the above-mentioned plane pei"pendiculai-
to AD and round the centres E and F. Now this is pre-
cisely the form in which they have been cut. Those por-
tions of the bevil teeth which lie in the conical surfaces
DPML and DIXY will therefore drive one another truly at
the instant when they pass through the line LY ; and there-
fore they will drive one another truly throngh an exceedingly
small distance on either side of that line. Now it is only
through an exceedingly small distance on either side of that
line niat any two given teeth remain in contact with one
another. Thus, then, it follows that those portions of the
teeth wliich lie in the conical surfaces Dll and DX work
truly with one another.

Now conceive the faces of the teeth to be intersected by an
infinity of conical surfaces parallel and similar to DM and
DS ; precisely in the same way it may be shown that those
portions of the teeth which lie in each of this infinite num-
ber of conical surfaces work truly with one another ; whence
it follows tliat tlie whole surfaces of tlie teeth, consh-ucted as
above, work truly together.

231. The modtjlus of a system of two conical oa

BEVn. WHEKLS.

Let the pressure P, and P^ be applied to tlie conical

./Google

OF TWO CONICAL

■wheels represented in the accompanying figure at perpen-
dicular distances a, and o, from tneir axes CB and OG ; let
the lengtli AF of their teetli be represented hy h ; let the
distance of any point in this line from F be represented by
X, and conceive it to be divided into an exceedingly great
number of equal parte, each represented by ^^x. Through
each of these points of division imagine planes to be drawn

perpendicular to the axes CB and CG of the wheels, dividing
the whole of each wheel into elements or lauiiuee of equal
thickness ; and let the pressures P, and P, be conceived to be
equally distributed to these laminae. The pressure thus dis-
tributed to each will then be represented by -^ a* on the'

the two pressures thus applied to the extreme laminte AH
and AK of the wheels, and let them be in equilibrium when
thus applied to those sections separately and independently
of the rest ; then if K represent the pressure sustamed along
that narrow portion of the surface of contact of the teeth ot'
the wheels which is included within these laminte,. and if K,,
and E, represent the resolved parts of the pressure K in the
directions of the planes AH and AK of mesa laminae;, the
pressnresj?, and E, applied to the circle AH are pressures
m equilibrinm, as also the pressvires p^ and R, applied to the
circle AK, If, therefore, we represent as before (Art. 216.)
by m, and m„ the perpendiculars from B and 6 upon the
19

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290 THE MODCLUS OF A STBTEM

directions of R^ and E„ and by L, and L„ the distances be-
tween the feet of tlie perpendicular «,, m, and a„ m, we
have (equation 236, 237), neglecting the weighta of tte
wheels,

P,~ — \ in, + (—') sin. 10, i E,

Pi and p, representing the radii of the axes of the two wheels,
and ipi and ?, the corresponding limiting angles of resistance.
Let 7j and 7, represent the inclinations of the direction of R
to the planes of AH and AK respectively ; then

Ri= E COS. Y,i ^1= R cos. y,.

Now it has been shown in the preceding article, that the
action of that part of tlie eiu'face of contact of the teeth which
is included in each of the laminse AH, AK, is identical with
the action of teeth of the same form and pitch upon two
cylindrical wheels AD and AL of the same small thickness,
situated in a plane EAD perpendicular to AC, and having
their centres in the intersections, 5 and y, with that plane of
the axes OB and CG produced. The reciprocal pressure R
of the teeth of the element has therefore its direction in the
plane EAD ; and if its direction coincided with the line of
centra DL of the two circles EA and AD, then would its
inclinations to the planes of AH and AK be represented by
DAH and LAK, or by ACB and ACG.

The direction of R is however, in every case, inclined to
the line of centres at a certain angle, which has been shown
(Art. 216.) to be represented in every position of the teetJi,
after the point of contact has passed the line of centres by
(fl+ip) ; wnere 0 represents the incHnation to AL of the line
^ which is drawn from the point of contact A of the pitch
icircles to the point of contact of the teeth, and where <p repre-
sents the limiting angle of resistance between the surfaces of
the teeth. To determine the inclination 7, of RA to the
plane of the circle AH, its inclination RAD to the line of
centres being thus represented by (fl+'p), and the inclination
of the plane AD, in which it acts, to the plane AH being
DAH, which is equal to ACB, let tliis last angle be repre-

,y Google

OF TWO CONICAL WHEELS. 291

sented "by i, ; and let Ats in the accom-
panving iigure represent tke intersection
of the planee AD and AH ; Aard repre-
senting a portion of the foitner plane and
Aach of the latter. Let moreover Ar
represent the direction of the pressure E
in the former plime and let Ad and AA be portions of the
lines AD and AH of the preceding figure. Draw ra per-
pendicular to the plane Aach^ and rd and ch parallel to Aoj,
and join dh: then rAo represents the inclination y, of the
direction of R to the plane AD, dAr represents the inclina-
tion {^+S) of AE to AH, and dAh represents the inclination
\ of the planes AD and AH to one another. Also, since Act
is perpendicular to the plane Ahd, therefore dr is pei-pen-
dicular to that plane,

,'. re = Ar sin. 7j = A(f see. (fl+(p)Bin. /,.

Also hd = Ad sin. ij, hut re — hd,

:. Adsec. (fl+v) sin, y,=:Adsm. i,;
.*■ sin. y, =co8. (!' + *>) sin, i,.

In hke manner it may be shown that sin. y, = cos. {6 +ip)
sin. („ (, being taken to represent the inclination KAL of the
planes AE and AK, which angle is also equal to the angle
ACG.

From the above equations it follows that

R,=E cos. 7,=R 4/1 — COS. '(fl -f

Rj=E cos. 7j=E yi — COS. X^+'P) sin. 'ij i

From the centre h of the circle AD draw hm perpendicular
to RA, then is BM (the pernendiculaa- let fall from the
centre of the circle AH upon the direction of E,) the preyeo
tion of hm upon the plane of the circle AH. To determine
the inclination of hn to the plane AH, draw An parallel to
bm ; the sine of the inclination of An to the plane AH is
tlien determined to be cos. DA«. . sin. 1,, precisely as the sine
of the inclination of Am to the same plane was before deter-
mined to be cos. DAm . sin. ij.

H'owDA)i=A!wt = s ~DAR=:o — (fl + ip); therefore the

sine of the inclination of An, and therefore of hm^ to the plane

./Google

THE MODCLDS OF A SYSTEM

AH is represented by the formula sin, (fl+y) sin, ij, and tha
cosine of its inclination by yT, — sin. '{i + (p) sin. \ ;
.'. m,=BM!=:5m j/1— Bin.'(S+?)8in.'i,.

Now it has been shown (Art. 216.) that the perpendicular
hm let fall from the centre of a spur wheel upon the direc-
tion of the pressure upon its teetli is, in any position of their
point of contact, represented (ecLuation 239) by the formula,

r, sin. (S + (p)+>^ sin. f,

where S, ?j, >■ represent the same quantities which they have
been taken to represent in this article ; but r^ represents the
radius iA of the circle AD, instead of the radius BA of the
circle AH; now SA=EA sec. DAH=r, sec.
this value for ?■, in the preceding formula, we have

hm=r, sin. ("+?) sec. i,+^ b'
.■. m, = jf, sin. (*+(p)sec. i,+>-sin. ?}
V'l — sin, \6 + (p) sin. 'i,.
Similarly it may be shown that

m.,= )r,siu. (^+9) sec. i,~>.sin.95

Vl^

.'{i + tfjein.'

Substituting the values of m, and m, above determined,
and also the values of li, and Rj (equations 268) in equations

./Google

OF TWO CONICAL WHEELS. 293

(267), and eliminating K between tliose equations, a relation
■will be determined between j), and p^ wliicli is applicable to
any distance of tbe pouit of^ contact of the teeth from the
line of centres,

let it now be aesmned that tlie number of the teeth of
the driven wheel is considerable, so that the angle ~ tra-
versed by the point of contact of each tooth may be small,
and the greatest value of the line \ the chord of an exceed-
ingly small arc of the pitch circle of the driven wheel. In

this case fl+? will very nearly equal - (Art. 220.) ; so that

COS. '(^+9) will be an exceedingly small quantity and may
be neglected, and sin. (8+9) very nearly equal wnity. Sub-
stituting these values in equations (268) and (269) we have

jn,,=r,4->.sin. 9 cos. i„ mj=rj— >■ sin. (pcce. 1,.

Substituting these values in equations (267) and dividing
those equations by one another so as to eliminate K,

ft «, ■

'

\«, }

'.

cos.

'.+

1 sin. ip.

■'■ ' " 1 — -— sin. © cos. I,— sm. o.

Whence performing actual division by the denominator of
the fraction, and neglecting terms involving dimensions
above the first in sin. 9, sin. ip„ sin. iji„

w ar ( /cos.', cos. i.\ . /p,L, \ .

Now if 4- represent the angle described by the driven
wheel or circle ELA, whilst any two teeth are in contact,
since >- is very nearJ.y a chord of that circle subtending this
small angle -f (Art. 230.); :.'K = r^^. Let * represent the

./Google

THE MODULirS OF i

a^le deacrilied "by the conical -wheel FK, whilst the circle
ELA describes the anj^le + i then, since the pitch circle of
the thin wheel AK and the circle ELA revolve in contact at
A, they describe equal arcs whilst they thus revolve, respec-
tively, through the unequal angles 4' and *. Moreover, the
radius A^ of the circle AL=AG sec. GA^=r, sec. i,, there-
fore 4'''j sec. jj^-tr,;

.■.■^=*cos. I (270).

Substituting the above valves of 4* ^^'d \ and observing

that — = -)

I * cos. I, sin. <f +

&\

■ (my

Multiplying both sides of this equation hjp, ^^, and ob-
serving that ^,ffl

ingly small angle described by the driving wheel AN, whilst
the driven wheel describes the angle A4-, so that if Aw, repre-
sent the work done by the pressure^, upon the lamina AH,
■whilst the angle a* is described by the driven wheel, then

i'A"

e have

— I 1 i' COS. ij sin. <p -f

— ) sin. 9, + (-^ sin. ipJ ;

or assuming a* infinitely small, and integrating between the
hmitsOand — {Art. 220.),

2*ffl,a, (, /cos. 1, COS. i,\
u,= --^■■- < 1 + * i 4— COS. I, Bin. f +

./Google

OF TWO CONICAL "WHEELS. ^90

Now the above relation tetween the work w, doiio by the
f ressure j?, upon the extreme element AH of the driving
wheel whilst any two teeth are in contact, and the pressure
«, opposed to the motion of the con-esponding element of
file driven wheel, ia evidently applicable to any other two
con-esponding elements ; the values of p„ *•„ r„ Lj and L,
proper to those elements being substituted in the formula.
If, therefore, we represent by liU, that increment of the
whole work Ui done upon the driving wheel, which is due
to any one of the elements into which we have imagined
that wheel to bo divided, and if we substitute for p, its

p
value -y^x, assign to Lj, L„ r„ r, their values proper to that

element, and represent those values by L, L', r, r',

2'rP,a, ( ^ /COS. I, COS. ij
aU,= ~U+ir - — -+ ^ COS. I, sm. 9+

or assuming &x infinitely small, and integrating between the

limits 0 and J, and obsei-ving that PjCTj^ represents the

11,
whole work U, done upon the driven wheel under the con-
stant pressure P, during the contact of any two teeth,

^, _ /COS. I, COS. i,\
U,=U,H-* — -' + ^ COS. .,6in.<p +

b

. (272).

K'ow a+3j being taken to represent the distance of the
point of contact of any two such elements from 0, and a to
represent the distance OF, the radii r and r' of these ele-
ments are evidently (by similar triangles) represented by

\ and r, repre-
senting the radii of the extreme elements NF and OP, or of
the pitcli circles of the lesser extremities of the wheels.
Also afisuming, as we have done, the pressures E, and P.

./Google

THE MODUI,US OF A

to "be perpendicular to the lines BA,
. GA joining the centre of each ele-
ment with their point of contact A,
so that the points M and N (see Jig.
p. 293.) coincide with the point A
(see accompanying figure)* ; and re-
presenting the angles ABD and ACE
made by the perpendiculars DB and
CE with the Rno of centres by S, and
i, respeetiyely; observing also that AD'=BA'— 2BA. BB
COS.. ABD+BD^, so that (^) := 1 - 2 (|^) cos. ABD +

/iiiil , we have, substituting, in the second number of this

equation, for BA or r its value J'l j 1 -F - j

or expanding the binomials in this expression, observing

that - is an exceedingly small quantity, neglecting terms

a
involving powers of that quantity above the first, and
reducing,

^m-'--m-

. (3T3).

Now Lj representing the value of L when x=0, and fl re
maining constant,

...(|)(cos.,-|)=.-(^)-(5;):

" The cirelea in this figure repres
wMeh wheels hsTP been imagined ti
Banio plane. Their pianes inlfiraect in AH,

, Google

OF TWO CONICAL -WHEELS. 291

Let now the angle AD33, made in respect to the first ele-
ment of the driving wheel between the pei-pendicular BD or
a, and the chord AD or L, be represented by j?;, and let »;,
represent the corresponding angle in the driven wheel, then

L,°— 3L,a, COS. )), + ^i'^aS ■'■ (ir} ~

Substituting these values of (— ^j and 2 Ml 1 cos. S ^j

in equation (273) ;

Extracting the square root of the binomial, and neglecting
terms involving powers of - above the first,

L L, la,\ M a,\L,x 1

,^ ^ , p, ein. to, /"L , p, sin. u), ( L, ,i )

.-. (Equation 2ra)ll^^y-i«=!i-^,-^;~'~i-C08.,,f.

b

P.sm.ip, rL' p,sin.(p, (L, ,5 )

Similarly — r f ~, ax = 1 — — -^ - cos. n. ( .

Substituting these values in the modulus (equation 272),

TT TT ( , /cos. 1, cos. l\

U, =:U, |l + *l— ^H ^lco3.(,sin. (p +

./Google

298 THE MODUIUS OF A BTBTEM

Now let the angla BC&, or the inclination of the axes,
from one to the other of which motion is transferred by the
wheels, be represented by 2i; therefore i,-i-i,=2i. Also a
sin. ii=!', and a sin. \^r„

sin. *i, _ sin. % 1

1 1 COS.'l cos.'i

__— ^■A„£^!l!!?-_ /COB- 'i COS. \\ /C0S.1, COS.1,1

/cos. I, COS. lA /I COS. I, 1\
" \ n^ n^ I \ji, COS. 1, n] ^ ^' '"

/COS. I, ' COS. I

Maw 25!:-!i - COS. Ki(' -0^ „ 1-tan.iO.— Qtaii.. .
" COS. I, ~ COS. |'-i(',-gj^ l + taii.^{i.-gtan..'

n|_ sin. ii_Bin. 5i+f(ij~jj)j _ tan. i + taTi.-|(i,— i,)
^^^ n,~ sin. .,~ eiu. j'-i(^^^ " tan.i- tan.i(. -jj

.-. tan. \ (i,— I,) = -iZ3 tan. . ;

(ra, + n,) — (n, — »,) tan. °
~ (ra, + Wj) + («-, — Wj) tan. '

-1 «-0+«-n,>n.'.

./Google

OF TWO CONICAL WTIEF.LS.

W, TO,/

\~ + T" COS.'

Substituting in the preceding relation, between U, and U„

f -, ( /I 1^2 sin. 'j 1 . P.sin.-p,

U'= {H-*H^ + -)-— —[Bin.9+ ^— '

which is the modulus of the conical or IdgvII wheel, neglecting
the influence of the weight of the wheel.

If for COS. >?, and cos. »?, we substitute tlieir values (see
p. 297), we shall obtain by reduction

4aL,

■ (2W),

from which equation it is manifest that the most faronrablo
directions of the driving or working ]
determined by the equations

232, It is evident, that if the plane of the revolution
of such a wheel be vertical, the influence of its weight must
be very nearly the same as that of a cylindrical or spur

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300 THE MODULUS OT A SYSTEM OF TWO CONICAL WHEELS,

wheel of the same weight, haviiig a radius equal to the mean
radius of the conical wheel, and revolving also in a vertical
plane. If the axis of the wheel he not horizontal, its weight
nauet be resolved inte two pressures, one acting in the plane
of the wheel, and the other at right angles to it ; the latter is
eifoctive only on the extremity of the axis, where it is borne
as by a pivot, so that the work expended by reason of it may
be determined by Art. 1^6, and will be found to present
itself under the form of N, . 8, where N, is a constant and 8
the space described by the pitch circle of the wheel, whilst
the work U, is done. Tlie resolved weight in the plane
of the wheel must be substituted for the weight of the wheel
in eonation (247), which determines the value of N. Assum-
ing the value of N, this substitution being made, to be repre-
sented by N,, the whole of the second terra of the modulua
wiU thus present itself under the form (K, + ^,)S.

( /I 1\ 3 sin. 'i ) . p, sin.ip.
r < t 1_ — I _ - — — — y em . ffl -1- — :

■■ ^.= 1+'- L~ + :;r - ^r^ «'n. 9 4

(JJ"^+N",)S (276).

233. Comparing the modulus of a system of two conical
wheels with that of a system of two cylindrical wheels
(equation 253), it will be seen that the fractional excess
of the work U, lost by the friction of the latter over that
lost by the friction of the former is represented by the
formula

The first tenn of this expression is due to the friction of
the teeth of tlie wheels alone, as distinguished from the fric-
tion of their axes ; the latter is due exclusively to the friction
of the axes. Eoth terms are essentially positive, since %

and 1J5 are in every case less than -.

Thus, then, it appears that the loss of power due to the
friction of bevU wheels is (other things being the same)
essentially less than that due to the friction of spur wheels,
so that there is an economy of power in the substitution of

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THE MODliLOa OF A 1

301

a l)evil foi' a spur wheel wherever ancli substitution is prac-
ticable. This result is entirely consistent with the experience
of engineers, to whom it is well known that hevil wheek run
Ughter than spur wheels.

234. The MoDura's of a Tkais of WnEELS.

In a train of wheels such as that shown in the accompany-
ing figure, let the radii of their
pitch circles be represented in
order by r,, r,, r, . . . t„ begin-
ning from the driving wheel ;
and let tii represent the pei-pen-
dicular distance of the driving
pressure from the centre of that
wheel, and a^ that of the driven
pressure or resistance from the centre of the last wheel of the
train ; TJ, the work done npon the first wheel, u, the work
yielded by the second wheel to the third, «, that yielded by
the fourth to the fifth, &c., and TJ, the work yielded by the
last or n"' wheel npon tlie resistance, then is the relation be-
tween U, and «, determined by the modulus (equation 252),
it being observed that the point of application of the resist-
ance on the third wheel is its point of contact h with the
third wheel, so that in this case a,=r^.

Th^e substitutions being made, and Lj being taken to
represent the distance between the point 5 and the •pT<^e(^ion
of the point a npon the third wheel, we have

U,= 1-f

, T.,P,

I.f>,

sin.pjiis,-i-N*.S,.

To determine, in like manner, the relation between «, and
«3, or the modulus of the third and fourth wheels., let it be
observed that the work «, which drives the third wheel has
been considered to be done upon it at its point of contact 5
with the fourth ; so that in this case the distance between the
point of contact of the driving and driven wheels and the
foot of the perpendicular let fSl upon the driving pressure
from the centre of tlie driving wheel vanishes, and tne term

• See note p. 266.

./Google

302 THE MODULUS OF A TEAIN OF WIIEELa.

which involves the value of L^ representing that line disaj)-
peai's from the modnlua, whilst the perpendicular upon the
driving pressure from the centre ot the driving wheel be-
comes r,. Let it also be observed, that the work of the
fourth wheel is done at the point of contact c of the fifth and
sixth wheels, so that the perpendicular upon the direction of
that work from the axis of the driven wheel is r^. We shall
thus obtain for the modulus of the third and fourth wheels,

«= \ l + *(_^ — \ein.0 + — 5^8in.d), [w,+KS,.
( \n, nj r^^ ) ^ ^

In which expression L, represents the distance between the
point c and the projection of the point S upon the fifth
wheel.

In like manner it may be shown, that the modulus of the
fiftli and sixth wheels, or the relation between m, andw,, is

u,= \ 1+* (i + i)sin. 9+^sm. ip, } m.+N, . S, ;

and that of the seventh and eighth wheels, or the relation
between v-. and m„

*( — H — )sin. (p + ^^ sin. ©, [ -w^+N,. S, ;

and that, if the whole number of wheels be represented by
2p, or the number of pairs of wheels in the train by p, then
is the modulus of the last pair.

n. 9j,^i|u,+Np.Sp;

In which expressions the symbols N,, N,, N, . . . Np , are
taken to represent, in respect to the successivepairs of wheels
of the tram, the values of that function (equation 247),
which determines the friction due to the weights of those
wheels ; and each of the symbols Lj, !„ L^ . . . Lp , the dis-
tance between the point of contact of a corresponding pair
of wheels and the projection upon its plane of the point of
contact of the next preceding pair in the train ; whilst the
symbols n„ «.„ % . . . n^ , represent the numbers of teetli in
the wheels ; r^, *■„ r„ . . . r^, the radii of their pitch circles ;
and Sj, S5, Ss . . . Sp , the spaces described by their points ot

./Google

303

contact «, J, c, ifec. wHlst the work TJi is done upon the S.rai
wlieel of the train.

Let OS suppc«e the co-efBcients of u„ «„ M^ . . . U„ in these
moduli to be represented by (l + H^,)' (^+i^!)! (l+H-j) ■ • • ■
{1 4-1*0 ) ; they will then become

M,=(X-i-^l,)M.-l-N, .8„
&c.=&c.

Eliminating «„ n„ u, , . .Up, between these equations, we
shall obtain an equation of the form

u.={i+f^,)(i+i-.)a+i^B)---(i+f^i>)^.+N".s . . . (an),

■where

NS=IJ,S, + (l + i^,)NA + (l+f^.)(l+^)^»S,+

+ (1-I-^)(1 + ^,) .... {l+i^p-,)NpSp (378).

Now let it be observed, that the space described by the first
wheel, at distance unity from its centre, whilst the space S,

is described by its circumference, is represented by — , and

space described in t!ie same time by the foot of the per-
pendicular «„ or the space , through which the moving
pressure may be conceived to work during that time; so

that — = . AJbo let it be obeerved that the space de-

scribed by the third wheel, at distance unity from its centre,
ie the same with that described at the same distance from

its centre bv the second wheel, so that — =— linlike

''a ''a

manner that the spaces described at distances unity from
their centres by the fourth and fifth wheels are the same, so

that — =— ; and similarly, that —=:-!, &c.=^&c. ; and
finally, -— --=-^-.

./Google

THE MOnULua OF J

J OF WHEELS.

Multiplying the two first of these equations together, thcii
the three first, tlie fowr first, &c., and tranaposijig, we have

-S, 8,=!i

:'ls.

Sabstituting these values of S„ S^, &c. in equation (278),
and dividing by S, we have

or if we observe that the quantities (*i, f-,, fr^, are composed
of terms all of which are of one dimension in sin. ip, sin. <?,,
sin. ip„ &c. and that the quantities N„ N,, N„ &c. (equation
247) are all likewise of one dimension in those exceedingly
small quantities ; and if we neglect terms above the iirst
dimension in those quantities, then

(w.)^"-+ •••}■•• (2re)-

If in lilie manner we neglect in equation (277) terms of
more than one dimension i]i |j.„ f*s, f*,, &c. we have

L\=U + ^+f*» + f^=+ - ■ ■ +f^ySU,+'N; . S.

Now iJ-,= ii (— _(, —J sin. 9 -\

, Google

I OF WHEELS.

Substituting these values of f„ f^s, &c. iu the preceding
equation,

?^"sin.9,+ -^Bin.<p,+...^^^£±1^2±lain.^^^jXJ,+N.S.(280),

■whicli is a general expression for the modulus of a train of
any number of wheels.

235. The work Ui whicli must bo done upon the flrat
wheel of a train to yield a given amount TJ, at the last wheel,
exceeds the work 1J„ or, in other words, the work done upon
the driving point exceeds that yielded at the working point,
by a quantity which is represented by the expression

\ ")sin.ip,U,H

+ . . .. + -^^2±i^sin.^+i) U,+NS . . . (281).

In which expression the ^s* term represents the expenditure
of work due to the friction of the teeth,* and varies directly as
the work Uj, which is done by the machine. The second
tenn represents the expenditure of work due to the friction
of the axes of the wheels, and varies in like manner direcHy
as the work done. "Whilst the tM/rd term represents the
expenditure of work due to the weights of the \vheels of the
train, and is wholly independent of the work done, but only
upon the space 8, through wliich that work is done at the
point where the driving pressure is applied to the train.

./Google

FEICnON OF THE AXES OF t

236. The ea^enditure of work due to the friction of the teeth

The work expended upon tlie friction of the teetli is repre-
sented by the formula

whose value is evidently less as the factor sin. ip is less, or as
the coefficient of friction between the common surfacee of the
teeth 18 less ; and as the numhei^ of the teeth in the different
wheels which compose the train are greater. Tlie number
of teeth in any one wheel of the train may, in fact, be taken
so small, as to give this formula a considerable value as com-
pared with TJj, or to cause the expenditure of work upon the
friction of tlie teeth to amount to a considerable fraction of
the work yielded by the train : and the numbers of teeth
of two or more wheels of such a train might even be taken
so small as to cause the work expended upon their friction to
tqvM or to siwpass by any number of tim^ the work yielded
by the train at its working point. This will become the
more apparent if we consider that the surfaces of contact of
the teeth of wheels are for the most part free from unguent
after they have remained any considerable time in action, so
that the limiting angle of resistance assumes in most cases
It much greater value at the surfaces of the teeth of the
wheels than at tlieir axes. From this consideration the
importance of assigning the greatest possible number of
teeth to the wheels of a ti-ain individually and collectively
is apparent.

23t. The expenditure ofworJc due to the friction of the axes.
This expenditure is represented by the formula

1- ^. + ~y sin. ^, + . . . 4_^yfa+Vn.^+i)u, . . . (2S8),

forming the second tenu of formula 280. Now, evidently, the
value of this formula is less as tl-.e quantities sin. ipi, sin. <p„
&c. are less, or as the limiting angles of resistance between
the surfaces of tiie axes and their bearings aj'c less, or the

./Google

FRICTION OF THE J

5 OF A TIJAIN.

lubrication of the axes
1,P, L,p, Ljp,

more perfect
', &c. are less.

and it is less as the

Now, L, being the distance between the point of contact i
, of the third and fourth wheels

f"^ ^s , R andtheprojectionofthe point of

I ° i^J f\ I \ contact a of the first and second

' upon the plane of those wheels,
it follows that, genei'ally, L, is
least when the projection of a
falls on the same side of the ams
as the point S ;* and that it
is least of all when this line faUs on that side and in the line
joining the axis with the point 5 ; whilst it is greatest of all
when it falls in this line produced to the opposite side of the
axis. In the former case its value is represented by r,—T„
and in the latter by r, -\-r, ; so that, generally, the maximum
and minimum values of L, are represented hy the expression

^s±^j! and the maximum and minimum values of —21'- by

I— -J- — j 0,. And similarly it appears that the maximum and

minimum values of —5^ are represented byl — j^ — J pj ; and

so of the rest. So that the maximum and minimum values of
the work lost by the friction of the i
by the e

1 p, sin. 9, +

from which expression it is manifest, that in every case the
expenditure of work due to the friction of the axes is less as
the radii of the axes are less when compared with the radii
of the wheels ; beingwhoUy independent of actual dimensions
of these radii, but only upon the ratio or proportion of the
radius of each axis to that of ite correspondmg wheel : more-
important condition ie but a pardeukc case cf the general principle
■ in Art, 168. ; from whicli principle it follows, tliat the driving
presaure ow each wheel should be applied on tlie same ade of the axis aa the
driven pressure.

, Google

303 THE WEIGHTS OF THE WHEELS,

over, that this expenditure of work is the least when the
wheels of the train are so arranged, that the projection of the
point of contact of any pair upon the plane of the next
following pair shall lie m the line of centres of this last pair,
between their point of contact and the axis of the driving
wheel of the pair; whilst the expenditure is greatest when
this projection falls in that line but on the other side of the
axis. The difference of the expenditures of work on the
friction of the axes under these two different arrangements
of the train is represented by the formula

2 ^ — sm. (pj + -LL am. f + -Li sm. ipj + ±i sm. (p. + . . f U :
I ^1 **, fi *", )

which, in a train of a great number of wheels, may amount
to a considerable fraction of U, ; that fraction of Uj repre-
senting the amount of power which may be sacrificed by a
false aiTaugoment of the points of contact of tlie wheels.

238. The expenditv/re ofioorh due to the wights of th^ several
wheels of the tram.

The third and last term J^ . S of the expression (280) repre-
sents the expenditure of work due to tlie weights of the
several wheels of the train; of this term the factor X is
represented by an expression (equation 279), each of tlie
terms of which involves as a factor one of the quantities N„
Kj, N„ &c., whose general type or form is that given in
equation (247), it being observed that the direction of the
driving pressure on any pair of the wheels being supposed
that of a tangent to their point of contact ; the case is that
discussed in the note to page 266. The olJier factor of each
term of the expressioji (equation 279) for N, is a fi'action
having the product n, n^ . . . of the numbers of teeth in all
the preceding drivers of the train, except the first, for its
numerator, and the product «., , n, . n,, . . , of the numbers of
teeth, in the preceding foUov.-ei's of tlie train for its denomi-
nator ; so that if the train be one by which the motion is to
be accelerated, the numbers of teetli in the followers being
small as compared with those in the drivers, or if the multi-
plying power of the train be great, and if the quantities
N„ Nj, H"„ &c., be all ^positive ; tben is the expenditure of
work by reason of the weights of the wheels considerable, as

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MODULUB OF A TKAIN IN WHICH THE DEn'EHS AKE EQUAL. 309

compared with the whole expenditure. Since, moreover, the
coefficiente of N^, ~S„ N",, &c., in the expreesion for N {equa-
tion 279) increase rapidly in value, this expeiidit\ire of work
is the greatest in reepeet to those wheels of the train which
are farthest removedTfroni its first driving wheel : for which
reason, especially, it is advisahle to diminish the weights
of the wheels as they recede from the driving point of the
train, which may readily be done, since the strain upon each
h-6 wheel is leas, as tlie work is transferred to it under

a more rapid motion,

239, 2'/ie modulus of a tram m whieh all the drivers are

Xial to one another and all the followers, and in which
povnts of eontad: of the d/ri/oefs cmd followers a/re all
simUarly stPuated.

The numbers of teeth in the drivers of the train being in
this ease supposed ecinal, and also the radii of these wheels,
nj=n,=«.,=n,=&e., *',=r,=7',=r,=&c. The numbers of
teeth in the followers being also eq^ual, and also the radii of
the followers n,=:ii,=»5=&c., j-,=:)',=r,=&c,

K moreover, to simplify the investigation, the dri/oin^
work TJ, be supposed to be done upon the first wheel of the
^ train at a point situated in re-

spect to the point of contact a of
that wheel with its pinion pre-
cisely as that point of contact is
in respect to the point of contact
h of the next pair of wheels of
the trahi ; and if a similar sup-
position be made in respect to
the point at which the driven work 11, is done upon the last
pinion of the train, then, evidently, L,=Lj=Lj= . , . =Lj,,
and {see equation 24:7) K,=N,= . . . =Kp.

The modulus (equation 280) wiU become, these substitu-
tions being made in it, the axes being, moreover, supposed
all to be of the same dimensions and material, and equally
lubricated, and it being observed that the drivers and the
followei-s are each^ in number,

IT,= j H-^p|l + i\6in. (p-f-^^sin.?,lu,+NS . . . .(384),

wMch is the modulus required.

./Google

: KK3ISTAMCE.

Moreover, the value of E" (equation 279) will become bv
the like substitutions,

-.(^i

i+e + r +

The Train of le;

240. A tram of equal driving wheels and equal foUo'wers
ieinff required to yield at the last wheel of the train a
given a/mount of work TI„ v/nd&r a vdodty m times greaier
or less than thai under which the work TJ, which drwes the
train is done by the mmmig power imon the first wheel; it
is required to determins what should ie the number p of
foms of wheels in the train, so that the work TJ, expended
through a given space 8, m dri/ving it, may he a minimum..

Since the number of revolutio

byt
if the train is required to be a given mnltiple or part of the
Lumber of reYolutions made by the first ■wheel, which mul
tiple or part is represented by tft, therefore (equation 231),

=e)*

Substituting these values in the modulus (equation 284);
substituting, moreover, for N its value from equation (285)j
■we have

./Google

0,+N,(ii»-l)

It is evident that the question is solved by that value of jp
which renders tliia function a minimum, or which satisfies

the conditions -^ = 0 and -^-r > 0. The first condition

op dp

gives by the dift'erentiation of equation (286),

-j rn^ll sl^ — 1 1— sin. <p + -4^ sin. fA-\ — sin.<p>

. (387).

p\m^ — ly

This equation may be solved in respect to^, for any given
values of the other quantities which enter into it, hy approxi-
mation. If, being differentiated a second time, the above
expression represents a positive quantity when the value of
P (before determined) ia substituted in it, then does that
value satisfy both the conditions of a minimum, and sup-
plies, therefore, its solution to the problem.

If we suppose ip,^0 and N,=0, or, in othor words, if we
neglect the influence of the friotion of the axes and of the
weights of the wheds of the train upon tlie conditions of the
question, we shall obtain

whence by reduction,

log.E '■

1+m

* This formula »as given, bj the kte Mr. Davis Gilbert, in hia paper on the
" Progcessive improTemenla made in the effieienej' of steam engines in Corn-
wall," published in the Tranaaotions of the Royal Society for 1830. Towards
the conclusion of that paper, Mr. Gilbert has treated of the methods best
adapted for imparling great angular velocities, and, in connection vith that

, Google

THE INCLINED PLANE.

The Inclined Plane.

241. Let AB represent the surface of an inclined plane on
whicb is supported a body whose centre of gravity is C, and
its weight W , by means of a pressure acting m any direction,
and which may be supposed to be supplied by the tension of
a cord pacing over a pulley and carrying at its extremity
a weight.

Let OR represent the direction of the resultant of P and
"W. If the direction of this line be inclined to the perpen-
dieulai- ST to the surface of the plane, at an angle OST
equal to the limiting angle of resistance, on that side of ST
which is farthest irom the summit B of the plane (as in
fig. 1), the body will be upon the point of slipping v/pwa/rds:
and if it be indined to the perpendicular at an angle OST,

ence due to the weigMa of the itlieela and to the friction of their aiea. The
author has in tain endetivoured to follo"' out tlie condensed reasoning bj which
Mr. Gilbert has arrived at this remarkable reeult ; It supplies another example
of that rare aagaoitj which he was accustomed to bring to the discnsrfon of
questions of practical science. Mr. Gilbert has (^ven the foUotring examples
of the solution of the formula by the method of approiuoadon;— If m=120,
or if the velocity is to be increased by the train 120 times, then the value of p
giyen by the above formula, or the number of pairs of wheels which should
ompose the train, so that it nay work with a minimum re^stance, reference
leing had only to tlie friction of the mirfaoes of the teeth, is S-llS; and the value

((the factor p(m^+-l){equBtion 288), which being multiplied by — sin. * TJ,

I spresenta the work eipended on the friction of the surfaces of the teetii, ia in
this ease 17-9 ; whereas its value would, according to Mr. Gilbert, be 131 If the
velocity were got up by a single p^r of wheels. So that the work lost by the
friction of the teeth in the one case would only be one seventh part of that in
the other. In like manner Mr. Gilbert found, that if !rt=:100, then y should
equal 3'S; in which case the loss by friction of the teeth would amount to the
sirUi part only of the loss that would result from that cause if J)=l, or if the
required velocity were got np by one pair of wheels.

If m=40, theny=:S-83| with a gain of one third over a single pair.

Ifm=B-69, theny=l.

If »i=12-85, thenp=a.

IflBz=46-8, theny=:3,

Ifm=16e-4, thenp=4.

It ia evident that when p, in any of tlie above examples, appears under the
form of a fraction, the nearest whole number to it, must be taken in practice.
The influence of the weights of the wheels of the train, and that of the friction
of the aies, so greatly however modify these results, that although they are
fully sufficient to show the existence in every case of a certain number of
wheels, wMch being assigned to a train destined to produce a given accelera-
tion of motion shall cause that Wain to produce the required effect with the
least eipenditure of power, yet they do not In any ease determine correctly
what that number of wheels should be.

, Google

equal to the limiting angle of resistance, but on the side of
ST nearest to the summit B (as in jig. 2.), tlien the body will
be upon tlie point of slipping downwai-ds (Art. 5 38.); the
former condition corresponds to the supeiior and the latter
to the inferior state bordering npon motion (Art, 140.).

Now the resistance of the plane is ec[ual and opposite to
the resultant of P and "W ; let it be represented by K.

There are then three pressures P, W, and li in equili-
brium.

,,^ ,,, P sin. WOE

.-. (Art. 14.) ^^=-. — pTYD

^ ' W sm. POK.

Let ZPAC=i, ZOST=limB, / of resistance =(p, let «
represent the inclination PQB of the direction of P to the
surface of the plane, and draw OV perpendicular to AB ;
then,

\^fig. 1, "WOE=WOV-|-SOV=EAC-|-OST=.+?,
and POE=PQB+OSQ=PQB+^-OST=*+a-? ;
in fig. 2., W0E=W0V-S0V=BAG-0ST=<-9,
and POE=PQB + OSQ=PQB+''+OST=*+<)+?;

;.*W'OE=i+(p; and POE=

f («+?);

the upper or lower sign being taken according as the body
is upon the point of shding up the plane, as vti fig. 1, or
down the plane, as in fig. % Or if we suppose the angle ^
to be taken positively or negatively according as the body ia
on the point of slipping upwards or downwards ; then gene-

■ rally "WOE=.+<p POE=^+(i-ip);

./Google

THE IXCLfflED PLANE.

P sin.(i+9) _aiii. (i + ^) _

p^^^BJii 0+9) ^2gg,

If the direction of P be parallel to the plane, /PQB o
^=0 ; and the above relation becomea

If 1=0 the plane becomes horizontal (fig. 3)., and the i
lation between P and "W asanmea the foiin

If e=0, P="W" . tan. 9, as it ought (see Art. 138.).

If the an^e PQB or & (fig, 1.) be increased so as to be-
come v—i, PQ will assume the direction shown in fig. 4,
and the relation (equation 289), between P and W will be-

P^-

^ !iH4+i) (292).

cos.(fl+1')

The negative sign showing that the direction of P must,
in order tnat the body may slip up the plane, be opposite to
that assumed in fig. 1. ; or that it ranet he a pushing pres-
sure in the direction PO instead of a pulling pressure in the
direction OP.

If, however, the body be upon the point of slipping down
the plane, so that ip must be taken negatively ; and i^ more-
over, <p he greater than I, then sin. (1+9), will become sin.
(1— ip)=— sin. (ip— 1), so that P will in this case assume the
positive value

p^-W . -^^^- ,^f ~'| (293),

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THE MOVKABLE INCLINED PLANE,

31o

■whicli determines the force just necessary under these cir
cumstancea to pull the body down the plane.

If i=!p, P=0, the body will therefore, in this ease, be upon
the point of slipping down the plane without the application
of any pressure whatever to cause it to do so, other than its
own weight. The plane is under these circumstances, said
to be inclined at the angle of repose, which angle is there-
fore equal to the limiting angle of resistance.

2i3. The direction of hast traotion.

Of the infinite number of different directions in which the
pressure P may be applied, each requiring a different amount
to be given to that pressure, so as to cause tlie body to slide
up the plane, that direction will require the least value to be
assigned to P for this purpose, or will be the direction of
least traction, which gives to the denominator of the fraction
in equation (289) its greatest value, or
which makes ^—9—0 or fl=?. The di-
rection of P is therefore that of least
traction when the angle PQB is equal to
the limiting angle, a relation which ob-
tains in respect to each of the cases dis-
cussed in the preceding article,

243. The Moveable Ixolined Plane.

Let ABC represent an inclined

plane, to the back AG of which

is applied a given pressure P^

and which is moveable between

the two resisting surfaces GH and

KL, of which eiuier remains fixed,

and the other is upon the point

of yielding to the pressure of the

" ■" plane.

If we suppose the resultants of the resistances upon the

different points of tlie two surfaces AB and BC of the plane

to be represented by R, and E, respectively, it is evident

that the directions oi these resistances and of the pressure P,

./Google

■will meet, wheu produced, in the same point O* ; and that,
since the plane is upon tiie point of slipping upon each of
the sui-faces, the direction of each of these reeietances is
inclined to the perpendicular to the surface of tlie plane, at
the point where it intersectB it, at an angle equal to the cor-
respouding limiting an^e of reeiBtance.

So lihat if ET and TS represent perpendicnlars to the
sui-farcee AB and BO of the plane at the points E and Y and
?„ HJ„ the limiting angles of I'esistance between these surfaces
of the plane ana the resisting surfaces GH and KL re-
spectively, tlien E,ET=(p„ R^S=ip,.

Now the pressures P„ E„ E, being in equiUbrium (Art.
W),

P, sin. EOF P, sin. EOF

H;^sin.DOF' "' B "sin. DOB'

But tlie foEP angles of the quadrilateral figure BEOP
being equal to four right angles (Euc. l-3ii), EOE— Sir —

EBF-OEB-OFB; but EBF=., OEB=j+(i, OFB=

j+f,. .■.E0F=.-i-9,-»,.

Similarly, D0E=2Tr-AB0-AE0-DAE ; but ADO =

.•.DOE=^+. + »,.

Since, moreover, DO is parallel to BO, both being per-
pendicular to AC, .•.DOF=«-OF0; but 010=—?,:

.I>0F=^+9..

.P. sin. |,_(, + (,, + ,,)j

sin. {<+<P,+1>,),
~ cos. 91 '

. p psin. (■ + ?, + '?,•)

(294.)

P, sin. |*-(j + ?,+90{
-■- sin.g + ...)

sin. (' + ?, + '?,).

COS. (l + ?,) '

" Since either is equal and opposite to the i

■esultantof the other i

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A BTSTEM OF TWO MOVEiBLE ISCLISKD PLANES. 317

,p^Il /jM^tMli,) (295.)

COS. (iH-9,)

111 the case in which the surface GH yields to the pressure
of tie plane, KL remaining fixed, we obtain (equation 131.)
for the modulus {see Art. 148.) observing that P,('')=R, sin. .
{equation 294),

-U_-Usin. (i + !P,+9,)

, (296).

In the case in which the sin-face KL yields, CH remaining
fixed, observing that P,W=E, tan. i (equation 295), we have

riB (l + T, + 9.) (2(,j),

cos.O + tPi)taii,i

Equations (296) and (297) may he placed respectively un-
der me forms

jj .mj^j+^)j„„,. (._+,,) + „„,, ,[

cos. 9, '

, ,, j-r cos. (?, + ?,)( tan. >+tan, ((p,+(p,) )

and 'J. = LJ, r^ 'i/-T z 'w ' — f •

em. ?i, ((cot. ?,— tan. i) tan. 1 )

The value of U, corresponding to a given value of U, is in

the former equation a tninimum when i^n, and in the latter

when

tan. 1= j J-. — ''^!-'^; , .-\ \ tan. (ip.+<p,) .... (298).
', ' sin. ?,Bm.{ip,+9,) ) \i w \ J

From the foi-mer of these equations it follows, that the work
lost by friction (when the driving surface of tlie plane is ita
hypotenuse) is less as the inclination of the plane is greater,
or as its mechanical advantage is less.

244, A nystem of two moveaVls inoUned planes.
let A and B represent two inclined planes, of which A

./Google

1 4

.1

ig|

i \ <

3tS A SYSTEM OF TWO MOVEABLE INCLINED

J, vests npfni a horizontal surface, and

[ ' receives a horizontal motion from

the action of the pressure P, ; com-
mnnicating to B a uiotion ■which ia
restricted to a vertical dkection by
the resistance of the obstacle D,
which vertical motion of the plane
__ is opposed by the pressure P, ap-

** plied to its superior surface. It is

required to determine a relation between the pressures P,
and P„ in their state bordering upon motion ; and the mo-
dulus of the machine.

Let H, represent the pressure of the plane A upon the
plane B, or the resistance of the latter plane upon the former,
and K, the resistance of the obstacle D upon the back of the
plane B ; then is the relation between E, and P, determined
by equation (394). And since R,, li^, P, are pressures in
equilibrium, the relation between 'R, and P, is expressed

(Art, 14.) by the relation=r= ■ ' -n'^-iv- 2fow E,Q is
"^ J ■^ p^ Bin. K,QK, '^

inclined to a perpendicular to the back of tlie plane B, at an

angle equal to the limiting angle of resistance between the

surface of that plane and tlie obstacle T) on which it is upon

the point of sliding. Let this angle be represented by 9,,

then is the inclination of R, to the back of the plane or P^Q

represented by^— 9, ; so that PsQRt=q— 9,-

And if "RjQ be produced so as to meet the surface of tlie
plane A in V, and YS be di'awn horizontally, liiQE,^

Qyit,+TE,Q=E,VS + SYA+TE.Q=?, + .+^+fl>„

where I represents the incJination of the superior surface of
the plane A or the inferior surface of the plane B to the
horizon. Substituting these values of P^QEj and E,QR, we
obtain

sin.( +i+ip, + 9,j

Kultiplying this equation by equation (294), and solving b
respect to P„

./Google

OF TIJEEE ISCIJKED TLAUES.

P _p Bill. (t + lp,+lp.) COS.!?
' 'cob. (t+ip,+ip,)cOe.(f

/.(Art. 152.) U.=;U,

__-,T sin. ((+<p, + tp,)eo8. 1

COS. (t+^j+9,) tan. ( cos.?.

A sT/siem of three inclined planes, two of wMeh a
ble, and the third Jlxed.

245. The inclined plane A, in the accompanying figure, is

,^, fixed in position, the plane B is

moveable upon A, having its upper

surface inclined to the hoi-izon at a.

less angle than the lower ; and 0 is

an inclined plane resting upon B,

which is prevented from moving'

horizontally by the obstacle D, but

may be made to slide along this

obstacle vertically. It is required

to determine a relation between

P, and P„ applied, as shown in the figure, when the system

is in the state bordering upon motion.

Let Rj, R„ Ej represent the resistances of the surfaces on
which motion takes place, <?, % <p, their limiting aueles of
resistance respecLtvely, andt^, t, the inclinations of the two
Burfacee of contact of B to the horizon. Since P„ K„ E^ are
pressures in equilibrium, as also Pj, Rj, R,

. P. sin.E,OE. R, sin. P,QR,
'■ R,- sin. P,OE,' P,~ sin. R.QR,'

Multiplying these equations together,

P, _ Bin. E,OR. . sin. F,QR,
P, -sin.P,OR,.ain.R,QR,

Draw OS and OT parallel to the faces of the plane B ; then
R.0E.=E,0S + Q0T-TO8; but R,OS= ^ —<?„ since OS ia

parallel to the inferior face of tlie plane B, alaoQOT=-— ip„
since OT is parallel to the snpei-ior face of the plane B ; and

./Google

330 A BT8TEM OP THKEE INCLINED PLANES.

TOS = the inclination of the facea of the plane B to one
another^i, — ly

.■.S,0E.= (2-»,)+ (^ -»,)-('.-'.)=»-(», + »,)-(',— ,).

AIbo P,QE,=|-E,QM=|-(.,.

Let P,0 be prodnced to V ; therefore P,OE,=* — E,OT=

»-(E,OS-SOV)=<- j g -9.) -., I = ^ + ., + q., Lastlj

Il,QE, = OQM+MQE,. ITow, jMQE,=p,; also, OQM =

,-QOT=--CQOT+TOV)=»- I g^».) +., j =g-.,+l.„

.-. E,QE.= ^ -,,+», + »,= I -(.,-(.,-?,).
^ 8in. |.-(o>. + »,)-(i -i,)}8in.(^-,,)

sin. -+., + •, .«in. ),-(.,-»,-»,)[

.(301).

. p _p Bin.i(<'. + l'.) + ('-'.)i°°s-t'. ^
" ' °Gos.{i, + 1ii)cos. )',— (9,+?,)! '

Wlienco we obtain for the modulus (Art. 152.), observing
_ sin. (',"!.)

„ _„ sin. (o,+?,+c,— tj)cos.(,COB. tjCOB^,
'~ ' cos.(»,— ?s— ?,) cos.(t, +p,) Bin. (*,— (^ ■

, Google

THE WEDGE I3KIVEN EY PEESSUk:

The Wedge dkiven bt Peessuee.

246. Let ACB represent an isosceles wedge, whose angle
AC13 19 represented by 2*, and which is
driven between the two resisting surfaces
DE and DF, by the pressure P,. Let E,
and K, represent the resistances of these
„.a»j surfaces upon the acting surfaces CA and
•■'■" y\ \ I t\^~"'^ ^^ ^^ *^^ wedge when it is upon the
■<\ \ I ^^ point of moving forwards. Then are the
T 1 y " directions of K, and R, inclined respec-
tively to the perpendiealai- Gs and R#
to the faces CA and CB of the wedge, at
angles each eqnal to the limiting angle of
resistance 9. The pressures R, and R, are
therefore equally inclined to the axis
of the wedge, and to the direction of P„ whence it follows
that E,=R„ and therefore (Art. 13.) that P,=2E, cos. ^GOR.
Now, since CQOR is a qiiadrilateral figure, its four angles
are equal to four right angles ; therefore G0R=2*— GOR~

OGC-ORO. ButG0R=2(; OG0=ORC=^+'p:

.•.GOR=*~(2(+29)) .•.iGOR-3^_(( + 9).

.■.P.=2E,sin.(f+()) (303).

WSience it follows (equation 121) that the modulus of the
wedge is

u,=n,"°-''+'' (304).

sin. (
This equation may be placed under tlie form

TJ,=:TJ, jcot. ip+cot. t}sin. tp.

The work lost by reason of the friction of the wedge is
greater, therefore, as the angle of the wedge is less ; and
infinite for a finite value of ip, and an infinitely small valae
oft.

The cmgle of tha vjsdge.
24-7. Let the pressure P„ instead of being that just snffi.-

,y Google

eient to drive theweclee,'beno'wsnpi^.

to be that ■which is only just eufficient to
keep it in its place when driven. The two
surfaces of the wedge being, under these
circumstances, upon the point of sliding
( backwards upon those between which the
wedge is driven, at their points of contact
G and K, it is evident that the directions
of the resistances i,G and i^R upon those
points, must be inclined to the normals
sGc and tB. at angles, each equal to the
limiting angle of resistance, but measured
on the sides of those normals opposite to

those on which the resistances E,G and Eijli are applied.*
In order to adapt equation (303) to this case, we have

only then to give' to ip a negative value in that equation. It

vnl) thei '

P,=2R,sin.((— f).

. (305).

So long as ( is greater than 9, or the angle C of the wedgo
greater than twice the hmiting angle ot resistance, P, is
positive ; ■whence it follows that a certain press\ire acting in
the direction in which the wedge is driven, and represented
in amount by the above fonnSa, is, in this case, necessary
to beep the wedge from receding from any position into
which it has been driven. So that if, in diis case, the pres-
sure P, be wholly removed, or if its value become less than
that represented by the above formula, then the wedge will
recede from any position into which it has been driven, or
'it ■will be started. If t be less than ip, or the angle C of the
wedge less than twice the limiting angle of resistance, P,
will become negative ; so that, in this case, a pressure, oppo-
site in direction to that by which the wedge has been driven,
will have become necessary to cause it to recede from the
position into ■which it has been driven ; whence it follows,
that if the pressure P, be now wholly removed, the wedge
■will remain fixed in that position ; and, moreover, that it
■will still remain fixed, although a certain pressure be applied
to cause it to recede, provided that pressure do not exceed
the negative value of P„ determined by the formula.

* This will at once be apparent, if we consider that the direction of the
reanllant pressare upon the wedge at G must, in the one case, be saoh, that if
it acted alone, it wonld cause the surfaee of the wedge to dip dowuwatda on
the surface of the mass at that point, and in the other case upwards; and ihal
the resistance of the mass is in each case opposite to tWa resultant pressure.

, Google

TBE WBDGE DKIVEH BT IMPACfT.

It is this property of remaining fixed in any position into
whicli it is driven when the force which drives it is removed,
tliat characterises the wedge, and renders it superior to
every other implement driven by impact.

It is evidently, therefore, a pnnciple in the formation of a
wedge to be thus used, that its angle should be leas than
twice the limiting angle of resistance between the material
which forms its surface, and that of the mass into which it
is to be driven.

The "Wedge DKrvGK bt Impact.

248. The wedge is usually driven by the impinging of a
heavy body with a greater or less velocity upon its back, in
the direction of its axis. Let "W represent the weight of
such a body, and V its velocity, every element of it being
conceived to move with the same velocity. The work
accumnlated in this body, when it strikes the wedge, wiU

1 W
then be represented (Art. 66.) by - — T". Now the whole of

this work is done by it upon the wedge, and by the wedge
upon the resistances of the surfaces opposed to its motion ;
if the bodies are supposed to come to rest after the impact,
and if the influence of the elasticity and mutual compi-eesion
of the surfaces of the striking body and of the wedge are
neglected, and if no permanent compression of their surfaces

follows tlie impact.* .'. U, = r- .

2 g

" The mfluence of these elements on the result maj be deduced from the
priodplei about to be kid down in the ehapKr upon impact. It results from
these, that if the surfaces of the impinging body and the back of the wedge,
by -which the impact is givea aud reeeiyed, ba exceedingly hard, as compared
with the flucfaces between which the wedge is driven, then the mutual pressure
of the impinging surfaces will be eiceedingly great as compared with the
resistance opposed to the motion of the wadga. Now, this latter being
neglected, aa compared with the former, the worls received or gained by the
wedge from the impact of the hammer will be shown in the chapter upon
impact to be represented by U~r^) "i 1 — where W, represents the

weight of the hammer, W, the weight of the wedge, and e that measure of
tile elasticity whose value is unity when the elBstieity is perfect. Eqaadng
this eipression with the value of TJi (equation B04), and negleodng the effect
of the elaatidty and oompresBion of the aurfiiceB G and R, between which ths
wedge IB driven, we shall obtdn the appioiimation

U,-

(14^)iWi'W,V sin, i

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324 THE WED&E DHIVEN BY IMPACT.

Substituting this value of TJ^ in equation 304, and solving ie
respect to tj„ we have

3 g ain.(_( + ^)

by which equation the work TJ, yielded upon the resistancea
opposed to the motion of tlie wedge by the impact of a given
weight W with a given velocity V ib detenniHed ; or the
weight "W necessary to yield a given amount of work when
moving with a given velocity ; or, lastly, the velocity V with
which a. body of given weight must impinge to yield a given
amount of work.

If the wedge, instead of being isosceles, be of the form of

,. a right angled triangle, as shown

Xo° in the accompanying figure, the

/c /// relationbetweenthe work U, done

' ■•■', //^^ upon its back, and that yielded

^^^^%/^'\ upon the resistances opposed to

ii.3^^^" i^~~^T its motion at either of its faces, is

'j A, 7^ represented by equations (296)

/^ 8 °"^ and (29T). Supposing therefore

* this wedge, like tlie former, to be

driven by impact, substituting as before for TJi its value

which the face AB of the wedge is its driving surface

_1 WV= sin, t COS. ?, ,g|j^, ,

^'~2 g ■sin.(i+?.,+90 ^ '''

when the base BC of the wedge is its driving surface,

,r _1'WT' tan, tens. (t+^i) /g^g-,

'~2 g ' sill. (( + 9,+?,)

From this eipreaaion it follows, that the useful work ia the grcateat, othei
tliinga being the fiame, when the weight of the wedge is oquol to the weight
of the hummer, aiid "hen the striking aurfacea are hard metals, ao that thf
value of e may approach the nearest possible to unil^.

./Google

THE ME AH PKESSUKB (

.-

.^1 —

r-

i

~i?~^

<

\

<,

respect to Uj, we have

COS. ((+9,+?,') tan, I

349. If the power of the wedge
be applied by the intervention of
an inclined plane moveable in a
direction at right angles to the di-
rection of the impact*, m shown in
the aecompanving figure, then snb-
Btituting for U, in equation (300)
half the vis viva of the impinging
body, and solving, as before, in

sin. {i + f^ + tf,) c

If instead of the base of the
plane being parallel to the direc-
tion of impact, it be inclined to
it, as shown in the accompanying
figure, then, substituting as above
in equation 302, we have

_1WY' eofi^',-
~2 g ' sin. (ip,H

9,)cos.(t,+9.)6in.(ti— *,)

Oc

:, COS. tiCOS.(p,

The 1

: Peesstjke of Impact.

250. It is evident from equations 306, 307, 308, that, since,
whatever may be the weight of the impinging body or the
velocity of the impact, a certain finite amount of ivoric U, ia
yielded upon the resistances opposed to the motion of the
wedge ; there is in every such case a certain mean resistance
R overcome through a certain space S, in the direction in
which that resistance acts, which resistance and space are
such, that

IIS=U , and therefore E=-

If therefore the space S be exceedingly small as compared

; wodge ia applied for tlia

, Google

THE 8CKEW.

with U„ there "will he an exceedinglj great resistance It
overcome hy the impact through that srnall space, howevei"
slight the impact. From this fact the enormous amount of
the resistances which the wedge, when struck hy the ham-
mer, is made to overcome, is accoKiited for. The power of
thus subduing enormous resistances hy impact is not how-
ever peculiar to the wedge, it is common to all implements
of impact, and belongs to its nature ; its effects are rendered
permanent in the wedge by the property possessed by that
implement of retaining peiinaneiitly any position into which
it 18 driven between two resisting suifaces, and thereby op-
posing itself effectually to the tendency of those surfaces, by
reason of their elasticity, to recover their original form and
position. It is equally tnie of any the slightest dwect impact
of the hammer as of its impact applied through the wedge,
that it is sufficient to cause any finite resistance opposed to
it to yield through a certain finite space, however gi'eat that
resistance may be. The difference lies in this, that the sur-
face yielding through this exceedingly small but finite space
under the blow of the hammer, immediately recovers iteelf
after the blow if the limits of elasticity he not passed ;
whereas the space which the wedge is, by such an impact,
made to travei'se, in the direction of its length, becomes a
permanent &

The Sceew.

251. Let the system of two moveable inclined planes re-

.^ presented in tig. p. 318. be formed of ex-

^^ .^..^ ceedingiy thin and pliable laminse, and con-

j ceive one of them, A for instance, to be
"' wound upon a convex cylindrical surface, as
shown in the accompanying figure, and the
other, B, upon a concave cylindrical surface
having an equal diameter, and the same axis
with the other ; then will the surfaces
EF and GH of these planes represent truly
the threads or helices of two screws, one of them of the form
called the male screw, and the other the female screw. Let
the helix EF he continued, so as to foi-m more than one spire
or convolution of the thread ; if, then, the cylinder which
carries this helix he made to revolve upon its axis by the
action of a pressure Pi applied to its circumference, and the
cylinder which carries the helix GH he prevented from re-

./Google

THE SCREW. 6£l

volving upon its axis by the opposition of an obstacle D,
which leaves that cylinder nevertheless free to move in a
direction parallel to its axis, it is evident that the helix EF
will be made to slide beneath GH, and the cylinder which
carries the latter hehx to traverse longitudinally ; moreover,
that the conditions of this mntual action of the helical sur-
faces EF and GH will be prec^ely analogous to those of the
surfaces of contact of the two moveable mclined planes dis-
cussed in Art. 244. So that the conditions of the equili-
brium of the pressures P, and P,in the state boi-dering upon
motion, and the modulus of the system, will be the same in
the one case as in the other ; with this single exception, that
the resistance Rj of the mass on which the plane A rests (see
fig. p. 318.) is not, in the case of the screw, applied only to
the thin edge of the base of the lamina A, but to the whole
extremity of the solid cylinder on which it is fixed, or to a
circular projection from that extremity serving it as a pivot.
Now if, in equation 299, we assume <p^=^0, we shall obtain
that relation of the pressures Pj and P, in liieir state border-
ing upon motion, which would obtain if there were no fric-
tion of the extremity of the cylinder on tiie mass on which it
rests 1 and observing that the pressure P, is precisely that
by which the pivot at the extremity of the cylinder is pressed
upon this mass, and therefore the moment (see Art. 176,
equation 183) of the resistance to the rotation of the cylinder

produced by the friction of tliis pivot by -P,ptan, <p^, where

p represents the radius of the pivot ; observing, moreover,
that the pressure which must be applied at the circumfe-
rence of the cylinder to overcome this resistance, above that
which would be required to give motion to the screw if there

were no such friction, ia represented by- P, -tan. <f„ r being

taken to represent the radius of the cylinder, we obtain for
the entire value of the pressure P, in the state bordering
upon motion

" COS. (i-ff.-t-fj) 3 V '

The pressure P, has here been supposed to be applied to
turn the screw at itBcwoumfer&noe; it is customary, however,
to apply it at some distance from its circumference by the
intervention of an ai-m. If a represent the length of such au

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328 THE SOEKW.

arm, measuring from tlie axis of the cylinder, it ia evident
that the preesuro P, applied to the extremity of that ai-m,
would produce at the circumference of the cylinder a pressure

represented hy P,-, which expression heing substituted for

P, in the preceding equation, and that equation solved in
respect to r„ we obtain finally for the relation between P.
and P, in their state bordering upon motion,

'\al I COS. {i+f,+f>,) 3\r I M ^ '

If in lite manner we assume in the modulus (equation 300)
9j=0, and thus determine a relation between the work done
at the driving point and that yielded at the working point,
on the supposition that no work is expended on the friction
of the pivot ; and if to the value of TJ, thus obtained we add
the work expended upon the resistance of the pivot which is
shown (equation 184) to be represented at each revolution

by yWpPj tan. ip^, and therefore during n revolutions by

-TnpPj, we shall obtain the following general expression for

the modulus ; the whole expenditure of work due to the
prejudicial resistances being taken into account.

-r- -IT sin. (t + iD,) COS. IJ- 4 „ .
' ' COS. (t-rip, + 9j tan. i ^3 ' '

Kepresenting byX the common distance between the threads
of the screw, *. e. the space which the nut B is made to
traverse at each revolution of the screw ; and obsei-v-

4 4 U,

ing that n>-F,=l\,, so tliat -■!<nfP,ts.i\.ii>,=-^^? tan. ?,=

2 2ffr P T> . , . -, . 2*7-

; ^. -.U, tan. 9,, m which expression— = cot. (, we

obtain finally for the modulus of the screw

U -U, i ^i"-('+^.)eos.. 2 p_ ^^^_ I ^^^_ ^ _ _ _ ^3^3^_
' ( COS. (i+-pj+9,) ^3r )

It is evidently immaterial to the result at what distance
from the axis the obstacle D is opposed to the revolution of

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APPLICATIONS OF THE 8CKBW. 329

that cylinder which carries tlie lamina B ; since the amount
of that resistance does not enter into the result as expressed
in the ahove formula, but only its direction determined by
the angle ?„ which angle depends upon the nature of the
resisting surfaces, and not upon the position of the resisting
point.

Applications of the Screw.

252. 'Ihe aecompanjing figure represents an application
of the screw under the circumstances described in the pre-
ceding ai'ticle, to the well known machine called the Vice.

AB is a solid cylinder carrying on its surface the. thread of a
male screw, and within the piece CD is a hollow cylindrical
surface, carrying the corresponding thread of a female
screw; this femSe screw is prevented from revolving with
the male screw by a groove in the piece CD, which carries
it, and which is received into a corresponding projection EF
of the solid frame of the machine, serving it as a guide ;
which guide nevertheless allows a longitudinal motion to
the piece CD. A projection from the frame of the instru-
ment at B, met by a pivot at the extremity of the male
screw, opposes itself to the tendency of that screw to tra-
verse in the direction of its length. The pressure P^to be
overcome is applied between the jaws H and K of the vice,
and the driving pressure Pi to an ann which cames round
with it the screw AB,

It is evident that, in the state bordering upon motion, the
resistance K upon the pivot at the extremity B of the screw
AB, resolved in a direction parallel to the length of that
screw, must be equal to the pressure P, (see Art. 16.) ; so
that if we imagine the piece CD to become flxed, and the

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330 APPLICAllOHB OF THE SOKEW.

piece BM to tecome moveable, being prevented from revolv
ing, as OD wa8,_ by the intervention of a gi-oove and guide,
t,heu raiglit tlie instmment be applied to overcome any given
resistance E opposed to the motion of this piece OD by the
constant pressure of its pivot upon that piece,
The screw is applied under these circumstances in the
common screw press. The piece
A, Hxed to the sohd frame of the
machine, contains a female screw
whose tliread corresponds to that
of the male screw ; this screw,
when made to turn by means of a
handle fixed across it, presses by
the intervention of a pivot B, at ita
extremity, upon the suiface of a
solid piece EF moveable verti-
cally, but prevented from turning
with the screw by grooves receiv-
i ing two vertical pieces, which
serve it as guides, and foi-m parts
, of the frame of the machine,
J The formulae determined iu
•^ Art. 251. for the preceding cas(
of the application of the screw, obtain also m this case, il
we assume ipj=0. The loss of power due to the friction of
the piece EF upon its guides will, however, in this calcu-
lation, be neglected ; tiat expenditure is in all cases exceed-
ingly small, the pressure upon the guides, whence their
friction results, bemg itself but the result of the friction of
the pivot E upon its bearings ; and the foi-mer friction being
therefore, in all cases, a quantity of two dimensions in
respect to the coefficient of friction.

if, instead of the lamina A (p. 836.) being fixed upon the
convex surface of a solid cylinder, and E upon the concave
surface of a hollow cylinder, the oiMieT be reversed, A being
fixed upon the hollow and B on the solid cylinder, it is evi-
dent that the conditions of the equilibrinm will remain fte
same, the male instead of the female screw being in this case
made to progress in the direction of its length. If, however,
the longitudinal motion of the male sci-ew B (p. 326.) be,
under tliese circumstances, arrested, and that screw thus
become fixed, whilst the obstacle opposed to the longitudinal
motion of the female screw A is removed, and that screw
thus becomes free to revolve upon the male screw, and also
to traverse it longitudinally, except in as far as the latter

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THE DIFFEKiiNTIAL SCEBW.

331

P-.A\

niotiori is opposed by a certain resistance

E, which the screw is intended, tinder

- tliese circumstances, to overcome ; then

■will the combination assnine the well

■ known form of the screw and nut.

■ To adapt the fonnulEe of Art. 251. to
this case, ?, mnat be made = 0, and

instead of asBiiming the friction npon the extremity of the
screw (equation 311) to be that of a solid pivot, we mnat
consider it as that of a hollow pivot, applying to it (b;^
exactly the same process as in Art. {251.), the formulie of
Art. (177.) instead of Art. (176.).

The Differential Sckew.

253. In the combination of three inclined planes d
in Art. 245., let the plane B be conceived of much greater
width than is given to it in the figure (p. 319.), and let it
then be conceived to be wrapped upon a convex cylindrical
surface. Its two edges ab and ed wUi thus become the
helices of two screws, having their threads of different incli-
nations wound round different portions of the same cyUnder,

•»CIfeaSK2S2?j

as represented in the accompanying figure, where the thread
of one screw is seen winding npon the surface of a solid
cylinder from A to 0, and the thread of another, having a
different inclination, from D to B.

Let, moreover, the planes A and C (p. 319.) he imagined
to be wrapped round two hollow cylindrical surfaces, of
equal diameters witli the above-mentioned solid cylinder,
and contained within the solid pieces E and F, through
which hollow cylinders AB passes. Two female screws will
thus be generated within the pieces E and Y, the helix of

, Google

333

HCNTER S BOKEW.

the OHO adapting itself to tliat of the male screw exter.diiig
from A to 0, and the helix of the other to that upon the
male screw extending from D to B. If, then, the piece E
be conceived to be iixed, and the piece F moveable in the
direction of tiie length of the screw, hut prevented from
turning with it by the intervention of a guide, and if a pres-
fiui'e Pi be applied at A to turn the screw AB, the action of
this coinbination will be precisely analogous to that of the
system of inclined planes dieeusaed in Art. 245., and the
conditions of the equihbrium precisely the same ; so that the
relation between the pressure P, applied to turn the screw
{when estimated at the circumference of the thread) and that
Pj, which it may be made to ovei'come, are determined by
equation (301), and its modulus by equation (302).

The invention of the differential screw has been claimed
by M. Prony, and by Mr. "White of Manchester. A com-
paratively small pressure may be made by means of it to
yield a pressure enormously greater in magnitude.* It
admits of numerous applications, and, among the rest, of
that suggested in the preceding engraving.

HnHTER's SCKEW.

the plane B (p. 319.) to be divided
b) a horizontal line, and the upper part
ti be wrapped upon the inner or concave
surface of a hollow cylinder, whilst the
iDwer part is wrapped upon the outer or
convex circumference of the same cylin-
der, thus generating the thread of a fe-
male screw within the cylinder, and a
male screw without it ; and if the plane
0 be then wrapped upon the convex sur-
^_ _ face of a solid cylinder just fitting the in-

side or conoive surtace of the above-mentioned hollow cylin-

• It will be sesn by reference to equation {301), that the -worTiiDg preesnie
Pa depends for its amount, not upon the actual incUnationa ii i, of the threads,
but oa the difference of thdr inclinations; so that ita amount may be enor-
mously increased by making the threads nearly of the same inclination. Thus,

neglecting friction, we have, by eiiuation (301), Fi=Pi ^/ ! _\- ; which

eipreasion becomes eicesdingly great when i, nearly equals la.

, Google

VARIABLE mCLIHATIOlT OF THE TKEEAD. 333

der, and the plane A upon a concave cylindrical anrface just
capable of receiving and adapting itself to the oviteide or
convex surface pf that cylinder, the mah screw thus generar
ted adapting itself to the thread of the screw within tlie hol-
low cylinder, and tlie female screw to the thread of that
without it ; if, moreover, the female screw last mentioned
be fixed, and the solid male screw be fi-ee to traverse in the
direction of its length, hut he prevented turning upon its
axis by the intervention of a guide ; if, lastly, a moving pres-
sure or power be applied to turn the hollow screw, ana a re-
sistance he opposed to the longitudinal motion of the solid
screw which is received into it ; then the combination will
be obtained, which is represented in the preceding engraving,
and which is well known ae Mr. Hunter a screw, navmg been
fii-st described by that gentleman in the seventeenth volume
of the Philosophie<d IramsaotWTis.

The theory of this screw is identical with that of the pre-
ceding, the relation of its driving and working pressures is
determined by equation (301), and its modulus by equation
(302).

The Theokt of the Sceew wrrn a Square Thbead dj ee-

FEEENCE TO THE VAKIABLE IxCLmATION OF THE ThEBAD AT

DiiTEKENT Distances skom the Axis.

255, In the preceding investigation, the inclined plane
which, being wound upon tho cylinder, generates the thread
of the screw, has been imagined to be an exceedingly thin
sheet, on which hypothesis every point in the thread may be
conceived to be situated at the same distance from the axis
of the screw ; and it is on this supposition that the relation
between the driving and working pressure in the screw and
its modulus have been determined.

now consider the actual ci^e m which the ihread

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ZZi VAKIABLE INCLIHATIOK OF THIS THKEAD.

of the screw is of finite tliickneee, and different elements of
it Bituafed at different distance from its axis.

Let mh represent a portion of the square thread of a screw,
in which foi-m of thread a line he, drawn from any point b on
tlie outer ed2;e of the thread perpendicular to the axis ef,
touches the thread throughout its whole depth M. Let AC
represent a plane perpendicular to its axis, and tt/' the pro-
jection of Sfi upon this plane. Take^ anj point in bd, and
let 5 be the projection of p. Let ep=r, mean radius of
thread =R, inclination of that helix of the thread whose
radius is E*=:I, inclination of the helix passing through p=i,
whole depth of thread =2D, distance between threads (or
pitch) of screw =L. Now, since the helix passing through
P may be considered to be generated by tlie enwrapping of
an inclined plane whose inclination is i upon a cylinder
whose radius is r, tlie base of which inclined plane will then
become the arc tq^ we have^j=^ . tan, i. But, if the angle
A/a be increased to Sir, pa will become equal to the com-
mon distance L between the threads of the screw, and tq
will become a complete circle, whose radius is r ; tberefore
L=2«r tan. t, and this being true for all values of r, there-
fore L=2*Il tan, I. Equating the second members of these
equations, and solving in respect to tan. *,

Etan. I .

tan, 1= (313).

From which expression it appears, that the inclination of the
thread of a square screw increases rapidly as we recede from
its edge and approach its axis, and would become a i-ight
angle if the tliread penetrated as fai- as the axis. Consider-
ing, therefore, the thread of the screw as made up of an in-
finite number of helices, the modulus of each one of which
is determined by equation (312), in terms of its cojTespond-
ing inclination t, it becomes a question of much practical im-
portance to determine, if the screw act upon the resistance
at one point only of its thread, at what distance from its axis
that point should be situated, and if its pressure be applied
at all the different points of the depth of its thread, as is
commonly the case, to determine how far the conditions of
its action are influenced by the different inclinations of the
thread at these different depths.

" Tbls may be ealled the mean lielii of the thraad. The term helii is here
taken to represent any spiral line drawn upon the surface of the thread; the
diatanee of every point in which, fcoja the axis of the screw, is the same.

, Google

OF THE THREAD. 835

"We shall omit the discussion of tlie former case, and pro-
ceed to tlie latter.

Let Pj represent the pressure parallel to its axis which is
to be overcome by the action of the screw. Now it is evi-
dent that the pressure thus pi-odnced upon the thread of the
screw is the same as though the whole central portion of it
within the thread were removed, or as though the whole
pressure Pj were applied to a ring whose thickness is A* or
2D, Now the area of this ring is represented by ^KE+D)'
— (R— D)°5 , or by 4«ED, So tliat me pressure of r^, upon

p
every square unit of it, is represented by tz-stt- I-et Ar

represent the exceedingly small thickness of such a ring
whose radius is r, and which may therefore be conceived to
represent the termination of the exceedingly thin cylindrical
surface passing through the point p ; the ai'ea of this ring is
then represented by SwMr, and therefore the pressure upon

. , P„ . 2«r^r 1 PoMt' „t .1 - ■ ■ 1 ^1 it
^^ "y . -p-p. , or by qWjY- -N ow this is evidently the

pressure sustained by that elementair portion of the thread
which passes through p, whose thickness is a?*, and which
may be conceived to be generated by the enwrapping of a
thin plane, whose inclination is t, upon a cylinder whose ra-
dius IS )■ ; whence it follows (by equation 311) that the ele-
mentary pressure aP^, which must be applied to tlie arm of
the screw to overcome this portion of the resistance P„ thus
applied parallel to the axis upon an element of the thread,
is represented by

.Ti lP„rAr\ jr\ | sin. (i-f is,) cos. «, „o ^ }

'^■=(*iy) (a) i -J.(.+,.+,.) +%'"■'■ 1 '

whence, passing to the limit and integrating, we liave

2RD»J I

T-'^'+fp^ ^'^^- '

COS.{i + fi, + %)

Now
6in.{i+ipj) cos.(p,_ tan. i+tan. ip,

COS. (i + Pi+iPj) ~~i— tan. (p, tan. ipj— tan. i (tan. i>,-t-tan. a,)

tan. i+tan. ip,

-(l--tan. <p^ tan. 9,) |l-tan. .tan. {'fi, + v7)} ^ 'P.+K'ii. <

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336 THE SCEEW WITH J

+taTi. (1',-l-T',) tan. °t. Keglecting dimensions of tan. 9, and
tan. fg above the first*,

:. ^■=QRo' / '^*^"' '''i + *^"- ' + *^'^- (^i+'Pt) tan.'i)»'' +

Iprtan.'pjf?)- (314).

Substituting ill tliis expression for tan. 1 its value (equation
813), it becomes

R + D

P /•

ri=;TWFr- / 5?-''tan.tt>,+Ertan.I+E'tan.''Itan.(>p, + ip,) +

R-D

\^r tan. ^^ dr.
Integrating and reducing,

^jtan.I+(l+ig)tan.^,^■|(^)tan.^,+

i;m.'Itan.(9,+?,)| (315);

whence M"e obtain by (equation 121) for the modulus,

n,=-ir,{i+|(i+ig)t...,.+|(i)t.n.,.+

tan.'Itan.(?..+(p,)|cot.ll (316).

256. "Wlience it follows that the best inclination of the
thread, in respect to the economy of power in the use of
the square screw, S& that which satisfies the equation

tan. I-

( tan.((F:,+<p3) )

The inclination of thread of a square screw rarely e
7°, so tliat the term tan. 'I tan. (9, + p,) rai-ely exceeds '015
t-an. (ipi+fps)! ^^<i ^fiy therefore be neglected, as compared

• The int«gration is readily effected without this omission ; and it miglit bo
derfrable so to effect it where the theory of wooden scrawa is under discussion,
the limiting angle of resiBtanoe being, in regpeot to Euoli screws, coiisiderable.
Tlie length and compHcntion of the resulting eiprea^on has caused llie omis'
Bioc of it in tlie teit.

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THE BlilAM OF THE STEAM ENGINE. 661

with thfi other temis of the expression ; as also may the
term i(— 1 tan. pi, eince the depth 2D of a eqiiare screw

being usually made equal to about -J^th of the diameter, this
terai doea not commonly exceed tu a tan. ?,,

Omitting these terms, observiug that L=2*E tan. I, and
eliminating tan. I,

=UJ l + j^("Rtan.9, + ||>tan.9j

.{SIT).
. (318).

The Beam of the Steam Enoine.

257. Let P„ P„ P„ P. represent the pressures applied by
the piefcon rod, the crank rod, tlie air pump rod, and the cold

water pump rod, to the beam of a steam engine ; and sup-
pose the directions of all these pressures to be vertical.*

Let the rods, by which the pressures P„ P,, P,, P^ are
applied to the beam, be moveable upon solid axes or gud-
geons, whose centres are «, d, h, e, situated in the same-
straight line passing through the centre C of the solid axis
of the beam.

Let p„ f„ p,, p, represent the radii of these gudgeons, p the
radios of the axis of the beam, and <?,, %, ij>,, ?„ ^ the limit-
ing angles of resistance of these axes respectively. Then, if
the beam be supposed in the state bordering upon motion

* A Bupposition whicii in no caae deviates greatty from the trutb, and any
error in which may be neglected, inaamuch as it een only influence the results
ftbout to be obtained in as fur as they have reference to the fiioUon of the
beam; so that any error in the result must be af two dimensions, at leaat, in
respect to the coefficient of friction and the small angle by whicli any preaanra
deviates from a vertical direction.

, Google

arfS THE BEAM OF THE STEAM ENGINE.

by tlie preponderance of P„ each gudgeon or axis being
upon the point of turning on its bearings, the directions of
tlie preeanres ?„ P„ P„ P„ E, will not be through the cen-
tres of their corresponding axes, but separated from them by
perpendicular distances a6\'erally represented by pi sin. ?„ p,
flin. <p„ p, sin, f,, p^ sin. (p^, and p siii. f, which distances, being
perpendicular to the directions of the pressui'ee, are all
measured horizontally.

Moreover, it is evident that the direction of the pressure
P, is on that side of the centre a of its axis which is nearest
to the centre of the beam, since the influence of the friction
of the axis a is to dimmish the effect of tliat pressure to turn
the beam. And for a like reason it is evident that the
■directions of the pressures P,j P„ P^ are farther from the
■centre of the beam than the centi-es of their several axes,
since the effect of the friction is, in respect to each of these
pressures, to increase the resistance which it opposes to the
rotation of the beam ; moreover, that the resistance It upon
the axis of the beam has its direction upon the same side of
the centre C as P„ since it is equal and opposite to the
resultant pressure upon the beam, and that resultant would,
by itself, turn the beam in the same direction as P, turns it.
Let now »,=Ca, a,=Cd, »,=;CS, ffl,=Ce. Draw the hori-
zontal line ofyCg', and let the angle aOf=:i. Let, moreover,
"W be taken to represent the weight of the beam, supposed
to act through the centre of its axis. Then since P„ P„ P„
P„ W, R are pressures in equilibrium, we have, by the
principle of the equality of moments, taking o ae the point
from which the moments are measured, P, . o/^=P, • og+
P, .oA+P, .'^+"W.^.

Now of^Q/"— Co=o, cos, fl— p, sin. <¥^—f sin, ip, og=Og-^
Co=:a, cos. ^ + pa siu. tPj+p sin. tp, oA=CA— Co=a, cos. ^ +
p5 sin. ipj— p sin. f, ok=Ck+Co=a^ cos. ^ + p. sin. (p, + p sin. >p.

.'.Pija, COS. *— (p, sin. ip,+p sin. 9)} =

P, \a, COS. a + (pt sin. % + p sin. if)\ + 1

P,{a,eo8.fl + (p,sin.(p,— p8in.ip)i+ V. ■ .(319).

P,iffi,co8.fl + (p,sin.(p,+psin.(p){ +W"psin.9 J

Multiplying this equation by fl, observing that a,6 repre-
sents the space described by the point of appheation of P„
so that P,ffi,'l represents the work TJ, of P, ; and similarly
that V,a,6 represents the work U, of P„ P.^^^, tliat V, of P„
and PjOJ,^, that IT, of P„ also that a^^ represent* tlie space S,

./Google

r THE STEAM EHGINE.

^p,6in.tp^ + p6in. ip\ I

- p sin. (p I

deserited tj the extremity of the piston rod very nearly ;

we have

.-, 1 , /f>,6in.<p, + f>sin.<])\ )
L,jcosJ-( ^- - )j-

uJcosJ+(^

uJcos.fl+-^

V,\cosJ+ (P>hMl^) I +W8.(i-).i

wliieh is the modulus of the beam.

Its form -will be greatly simplified if we aesiime cos. ^=1,
since 6 is small,* suppose the coefHcient of friction at each
axis to be the same, so that ip=9,=9,=(p,=:(p„ and divide by
£he coefficient of tJ„ omitting ternis above the first dimen-
sion in — sin, ip, &c. ; whence we obtain by reduction

TTilJ./tti.J.thfil

U.J1+ (^ + t±Pi|si„.f J +WS,(i)sm,f

258. The best position of the axis of the liea/in.

Let a he taken to represent the length of the beam, and x
the distance aG of the centre of its axis from the extremity
to which the driving pressure is applied.

• In practice the angle 9 nerer exceeds 20°, so that cos. 8 never differs from
unity by more than '060807. The error, resulting from which difference, \a
the friction, estimated as abore, must in all cases be inconsiderable.

, Google

340 THE BEA.M OF THE BTEAM ENGIHB.

Let the infiuGJice of the poeition of tlie axie on the
economy of the work necessary to open the valves, to work
the air-pump, and to overcome the friction produced by the
weight of the axis, he neglected ; and let it be assumed to
he that, by which a given amount of work U, may be
yielded per stroke upon the crank rod, by the least poseible
amount TJ, of work done npon the piston rod. If, then, in
equation (321), we assume the three last terms of the second
member to he represented by A, and observe that a^ in that
equation is represented by x, and a, by a~x, we shall
obtain

U,=

il±h.

p|TJ,-i-A.

The best position of the axis is determined by that value
of ce which renders this function a minimum ; which value
of a: is represented by the equation

, (323.)

If p.>p„ thenf^ — -] >1 and !S<.ia: in this case, there-
\p+p,/

fore, the axis is to be placed nearer to the driving than to
the working end of the beam. K p5<i'i, the axis is to be
fixed nearer to the working than to the driving end of the
beam.

259. It has already been shown (Art. 168.), that a
machine working, hke the beam of a steam engine, under
two given pressures about a fixed axis, is worked with the
greatest economy of power when both these preesurea are
applied on the same side of the axis. This principle is
manifestly violated in the beam engine ; it is observed in
the engine worked by Crowther's parallel motion,* and in
the marine engines recently introduced by Messi's. Seaward,
and known as the Gorgon engines. It is difficult indeed to
defend the use of the beam on any other legitimate ground
tlian this, that in some degree it aids the fly-wheel to
equalise the revolution of the crank arm,t an explanation

• As used in the mining districtJi of the north of England.

t The reader is referred to an admirable discussion of the equalising power
of the beam, by M. Coriolis, contained in the thirteenth volume of the Jmmial
de VEeole Polytechniqu^.

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THE CKANK.

341

which does not extend to its use in pumping engines,
where, nevertheless, it retains its place; adding to the
expense of construction, and, by its weight, greatly increas-
ing tiie prejudicial resistances opposed to the motion of the

iO. The modulus of the crank, the direction of the j
anod "be^/ng pwrallel to that of the d

Let CD represent the arm of the crank, and AD the con-
necting rod. And to simplify the
investigation, let the connecting
rod be supposed always to retain
its vertical position.* Suppose the
weight of the crank arm CD, act-
ing through its centre of gravity,
to he resolved into two other
weights (Art. 16), one of which "W,
is applied at the centre G of its axis
and the other at the centre e of
the axis which unites it with the
connecting rod. Let this latter
i weight, when added to the weight
of the oojmeciing rod, be repre-
sented by Wj. Let P, represent a
pressure opposed to the revolution
of the cranK, which would at any
instant be just sufficient to balance
the driving pressure P, transmitted through the connecting
rod; and to simplify the investigation, let us suppose the
direction of the pressure P, to be vertical and downwards.

Let Oc=a, CA,=«„ CA^^a^ cGW,=6, radii of axes 0
and c=f^, p„ lim./s of resistance=9„ ?„ 'W'=rwhole weight
of crank arm and connecting rod=W, + W,.

Since the crank arm is in the state bordering upon
motion, the perpendicular distance of the direction of the
— ■-'—"a upon its axis 0 from the centre of that axis, is

* Any error resulting from this hypotht
question only in as far as t!ia frictio
sion3 at least in terms of the coefficif
atioQ of the couneoting rod from the

affecting tlie conditions of the
id, and being of two dimen-
and the small angular devi.

, Google

repjesented by p, sin. if, (Art, 153.). Tlie resistance is iho
equiLltoP,±(P,-|-'W); PibeingsuppoaedgreatertbanPj+W^
and the sign ± being taken
according as the direction of
P, is downwards or upwards,
or according as the crank arm

serving that the c
according as the arm i

ascending arc. Whence it
follows, that the moment of
the resistance of the axis about
its centre is represented by
jP.±{P,+"W)| p, sin. ?,.
i^jfai\ ■^'^'^^ '■^^ pressures P„ P,, and

■" '' the resistance of the axis, are
pressures in equilibrium.
Tlierefore, by the principle of
the equality of moments, ob-

^iressure is represented by PiiW;,

.escending or ascending,

+ !P.±(P,+W)i p. sin.?,.

(P,±W,) a,=V,a

Since moreover the axis e, which imiteS the connecting
rod and the crank arm, is upon the point of turning upon
its bearings, the direction of the pressure P, is not through
the centre of that axis, but distant from it by a quantity
represented by p, sin. 9,, which distance is to be measured
on that side of the centre e which is nearest to C, since the
friction diminishes the effect of P, to turn the crank ann.

Substituting this value of tu, in the preceding equation,
(P.±"W,) (ffl sin. e—p, sin. fl=,)=P,«,+ |P,±(P,+

"W")} p, sin. ip, (324:).

Transposing and reducing

P, ja sin. fl— p, sin. pj—p, sin. ?,} =PJ»,±p, sin. ?,} ±
Wp, sin. ipj^'W,{a sin. ^— pj sin. (p,);

which is the relation between P, and P, in tlieir state bor-
dering upon motion. Now if Afl represent an exceedingly
small angle described by the crank arm, a^&i will represent
the space through which the resistance P, is overcome
whilst that angle is described, and P^a^Afl will represent tlie

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THE CEABK. 343

increment aU, of the work yielded by the crank whilst that
small angle is described. Multiplying the above equation
by KjAfl, we have
Tja^la sin. 6—f, sin. ip^— p, sin. 9,1'^^= {a,±?, sin. 9,}Air,±

■Wa,p, sin. tp,Afl^W,«, (a sin. 3— f, sin. (p,)Ad (325).

whence passing to the limit, integrating from ^=6 to &=
w— G, and dividmg by a,

P, {2acos.e— (*-26)(p,sin.(p, + p,aia.aj,)i = | li^sin.^, i U,±

■W(*— 3e)p,sin.(p,^"W,{2(;tcos.e— p,(*— 2e)sm.<p,§. . (326).

Kow, let it be observed that 2a cos. 9 represents the pro-
jection of the path of the point e upon the vertical direction
of P„ whilst the arm revolves between the positions 9 and
*— 9; so that P,2ffl cos. 6 represents (Art. 52.) the work
Ui done by P, upon the crank whilst the arm passes from
one of these positions to the other, or whilst the work U, is

yielded by the cranli. "Whence it follows that F^=:^ sec. 6,

Substituting this value of Fi, and reducing we obtain

U, 1 1- g-e)sec. e(-^ sin. <p,+^ sin. A I =

jl±^sin.?, ju, ±W('r-26)p, sin. (p,q:WJ2a cos. 0-

?, {^-20) sin. 9,i (327).

By which equation is determined the modulus of the crank
in respect to the descending or ascending stroke, according
as we take the upper or lower signs of the ambiguous terms.
Adding these two values of the modulus together, and
representmg by tJi the whole work of P„ and by TJ, the
whole work of *r„ whilst the crank arm makes a complete
revolution, also by u, the work of P, in the down stroke;
and Mj in the -wp stroke, we obtain

U. jl-f—ejsec.ef-Jsin. (p,+-^8in. 9,] i =U,^-
(M -«,) ^ sin. 9. (328),

which is the modulus of the crank in respect to a vertical

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34i THE CB4NK.

direction of the driving pre^ure and of the resistance, the
arm being supposed in each half revolution, first, to receive
the action of the driving pre^ure when at an inclination oi
0 to the vertical, and to yield it when it has again attained
the same inclination, bo as to revolve under the action of
the driving procure through the angle t— 2©.

In the double-acting engine, u^—u^=0 ; in the single-act-
ing engine u,—0. llie work expended by reason of the
friction of the crank is therefore less in the latter engine than
in the former, when the resistance P, is appKed, as shown
in the figure, on the side of the ascending arc.

If the ai-m sustain the action of the driving pressure oon-
stcmtly, 9=0, and the modulus becomes, for the dcmble-act-
mg engine,

( 3\a a I ) "

or, dividing hy the co-efficient of Ui and neglecting dimen-
sions above the first in sin. 9,, sin. (p„

u.=

in. (p,-|-~ sin. pj [U, . . . . {

The modulus not involving the symbol "W which repre-
sents the weight of the crank, it is evident that so long as P,
and P, ai"e vertical and P, gi'eater thanP^+W, the economy
of power in the use of the crank is not at all influenced by
its weight and that of the connecting rod, the friction being
upon the whole as much diminished by reason of that weight
in the ascending sti-oke as it is increased by it in the descend-
ing stroke.

It is evident, moi-eover, that if the friction produced by
the weight of the crank be neglected, the modulus above de-
duced, tor the case in which the directions of the pressures
P, and P, are vertical, applies to every ease in which the
directions of those pressures are parallel.

The condition P,>p5-f-'Wevidently obtains in every other
position of the crank aa-m, if it obtain in the hoiizontal position.

Kow, in this position, P,=— Pj, if we neglect friction. The

required condition obtains, therefore, if P,>— P, + W To

satisfy this condition, a, must be greater than ts, or the
resistance be applied at a perpendicular distance from the

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THE DEAD POINT IN THE OEANK. 345

axis greater than the length of cvaiik ami, and so much

greater, that Pi (l j may exceed W. These conditions

commonly obtain in the practical application of the crank.

261. Should it, however, be required to determine the mo-
dulus in the case in wliich P, is not, in every position of the
arm, greater than P^+W, let it be observed, that tliia condi-
tion does not affect the determination of the modulus (equa-
tion 327) in respect to the descending, but only the ascend-
ing sti'oke ; there being a certain petition of tlie arm as it
ascends iti which the r^ultant pressure upon the axis repre-
sented by the formula jP,—(P,-|-'W)!, passing through zero,
is afterwards represented by {(Pj-j-W)— P,j ; and wlien the
arm has still further ascended so as to be again inclined to the
vertical at the same augle, passes again through zei'o, and is
again represented by the same formida as before. The value
of this angle may be determined by substituting P, tor
Pj-t-V in equation (334), and solving that equation in re-
spect to S, Let it be represented byl, ; let equation (325)
be integrated in respect to the ascending stroke from 6=0
to 6=6^, the work of P, through this angle being represented
by u, ; let the signs of all the terms involving p, sin. ip, then
be changed, which is equivalent to changing the formiila re-
presenting the pressure upon the axis from tPi~"(Pa+^^5
to KPj-f-W)— P,f ; and let the equation then be integrated

frora^=a,tofl=^, the work of P, through this angle being re-

l by -u, ; 2(w,-f M,) will then represent the whole
i; Uj done by P, in tlie ascending arc. To determine
this sum, divide the first integral by tlie co-ef&cient of «„
and the second by that of w^, add the resulting equations,
and moltiply their sum by 2 ; the modnlns in respect to the
ascending arc will then be determined ; and if it be added
to the modulus in respect to the descending arc, the modu-
lus in respect to an entire revolution will be known.

The Dead Points in the Ckakk.

862. If equation (324) bo solved in respect to P, it be-
comes

./Google

THE rODBLB UKABK.

lain. 6— p, sin, fj— pi sin, tp, j "^

"Wp, 9ia.ip,— "Wi(g6in.d— p, sin, ip,)
dsin. fl~pjSJii. p,— p,sin.9,

In that position of the arm, therefore, in which

the diiYing pressure Pj neceseary to overcome any given re-
sistance P, opposed to the revolution of the crank, assnmea
an infinite value. This position from which no finite pres-
sure acting ill the direction of tlie length of the connecting
rod is sufficient to move the ai'm, when it is at rest in that
position, is called its dead point.

Since there are four values of S, which satisfy equation
(830) two in the descending and two in the ascending semi-
revolution of the arm, there are, on the whole, four dead
points of the crank.* The value of P, being, however, in all
cases exceedingly great between the two highest and the two
lowest of these positions, every position between the two
former and the two latter, and for some distance on either
side of these limits, is practically a dead point.

Toe Double Ckank.

263. To this crank, when apphed to the steam engine, are
affixed upon the same solid shaft, two arma at right angles
to one another, each of which sustains the pressiire of the
steam in a separate cylinder of the engine, wliich pressure is
transmitted to it, from the piston rod, oy the intervention of
a beam and connecting rod as in the marine engine, or a
guide and connecting rod as in tiie locomotive engine.

• It has been customary to reolion theoretically only two dead points of the
crank, one m its higiiest and the other in its lowest position. Every practical
man is acquaiated with the fallacy of this conclnsion.

, Google

THE DOUBLE C

847

In eitiiercase, the connedingroda
may te supposed to remain con-
stantly parallel to tlieraselves, and
the pressures applied to them in
different planes to act in the same
plane,* without materially affecting
the results about to be deaaced-f

Let the two arms of the crank be
supposed to be of the same length a ;
let the same driving pressure P, be
supposed to be appDed to each ; and
let the same notation be adopted in
other respects as was used in the
case of the crank with a single arm;
and, iii-st, let us consider the case
represented in fig. 1, in which both
ai'ms of the crank are upon the same
side of the centre 0.

Let the angle W,CB=d ; therefore W,CE=2+^ : whence

it follows by precisely the same reasoning as in Art, 360.,
that the perpendicular upon the direction of the driving
pressure applied by the connecting rod AB is represented
(see equation 323) by a sin, fl— p, sin. (p„ and the per-
pendicular upon the pressure applied by the rod ED by

"•(i-'l-

-pj sm. (pj, or a i

3s. fl— p5 sin. %. Let now <

be taken to represent the perpendicular distance from the
axis C, at which a single pressure, equal to 2P„ must be ap-
plied, 80 as to produce the same effect to turn the crank as
IS produced by the two pressures actually applied to it by
the two connecting rods ; then, by the principle of the equa-
lity of momenta,

2Pi(i,=Pi(» ein,^— p, sin.9,)-|-Pi(« cos. ^— p, sin. ip,) ;
.*. a,=-^o;(sin. fl-h COS. *)— pjsin. 9, ;

* This principle will be more fully disoussed by a reference to the theory of
statical couples. {See Pritchdrd on Statical Couples.)

t The relotire dimenaons of the crank arm and connecting rod are here sup-
posed to be those usually giren to these parts of the engine ; the supposition
does not obtain in the case of a short connecting rod.

, Google

Sis THE DOUBLE CEAHK.

O' I . . * . ■ *l

,-. a,= —r- (311. « COS. T + COS. a sm. 7I — p, sm. ?,=
;iBm.()+j)-f,»ii,.,,;

which expression becomes identical with the value of «„ de-
termined by equation (333), if in the latter equation a he
Teplaced by -^, and * by 3 +— Whence it follows that the

conditioDB of the equilibrium of the double crank in the
state bordering upon motion, and tlierefore the form of the
modulus, are, whilst both arms are on tlie same side of the
centre, precisely the same as thfffie of the single crank, the
direction of whose arm bisects the right an^e BCE, and
the length of whose arm equals the length of either ai'm of
the doable erank divided by |/2.

Now, if fl, be taken to represent the inclination W^CF of
this imaginaiy arm to W,C, both ai-nis will be found on the

8ame side of the centre, from that position in which ^, = j
to that in which it equals { * — ^r). If) therefore, we substi-
tute ~ for 0, in equations (326), and for o, — , and add these

i V2

equations together, the symbol 2 U, in the resulting equa-
tion will represent the whole work yielded by tlie working
pressure, whilst both aiins remain on the same side of the
centre, in the ascending and the descending arcs. We thus
obtain, representing the sum of the driving pressures upon
the two arras by Pj,

2P, |« - ^ (p, sin.ip, + P, Bin.9,) j = 2U, (331).*

Throughout the remaining two quadrants of the revolution
of the crank, the directions of the two equal and parallel
pressiu'os applied to it through the connecting rods being
opposite,' tlie resultant pressure upon the axis is represented
by (P, + W), instead of JP,±(P,-l;W)f ; whilst, in other
respects, the conditions of the equilibrium of the state bor

' Whewell'a Mechanics, p. 26.

, Google

THE DOUBLK CRANK.

denng iipon motion reraiiin the same as before ; that is, the

^. 2 same as though the pressure Pi were

apphed to an imaginary arm, whose

length if

fa

, and ■v

e position co-

incides with GF. i^"ow, referring to
equation (324), it is apparent mat
this condition will be satisfied if, in
... that equation, the ambiguous sign of
h \ (P!i+ W) be suppressed, and the
value of Pi in the second member,
"1 which 18 multiplied by p, sin. f^, be
assumed =0 ; by which assumption
tlie term —p^ sin. (f^ will be made to
disappear from the left-hand member
of equation (335), and the ambiguous
signs which affect the first and second
terms of the right-hand member will become positive. !Now,
these substitutions being made, and the equation being then

integrated, first, between the limits 0 and -, and then be-
tween the limits --- and *, the symbol TJ, in it will evidently

represent the work done during each of those portions of a
semi-revolution of the imaginary arm in which the two real
ai'ms of tile crank are not on the same side of the centre.
Moreover, the integral of that equation between the limits 0

and 7' is evidently the same with its integral between the

limits -r and ir. Taking, therefore, twice the former inte-
gi'al, we have

2P.aJ-^(l~cos.^)-7f'=8in.?. I = L,+ p,6in.^i

2U,+2 W«,p, sin. p,T2W,^. j — (l-cos.j) -^ p, sin. ?, j

Dividing this equation by (Uj-]- p, sin. 9,), or by a,
1 IH — ^ sin. 9, l; and neglecting terms above the first dimen-
sion in sin. ra, and sin. <:>.„

./Google

THE DOCBLK CKANK.

sm.f, l=2ir, +
^W,,.m.,,T2W.|-|(l-c„4)(l-^,m.,.)-

ill -which equation SU, represents the work done in the
descending or ascending ai'cs of the imaginary arm, accord-
ing as the ambiguoua sign is taken positively or negatively.
Taking, therefore, the sum of the two values of the equation
given by the ambiguous sign, and Representing by 4U, the
whole work done in the descending and ascending arcs, dur-
ing those portions of each complete revolution when toth of
the arms are not on the same side of the centre, we have

4TJ5+'Wirpj sin. (Pj ;

2P, ] (((4/2— 1)— a{ V2— 1) ^ sin. ip,— 2 p, sin. <p, \ =

iUj-h'W'irfi, sin. 9,.

Adding this equation to equation (331), and representing by
IT, the entire work yielded duiing a complete revolution of
the imaginary arm,

2P, |of3 — o(y2 — 1)— fiin.ip, — ^(2p,sin.p,+p.sin.9,) [
=U,-|-"VT^P,Bin.9,.

But if U, represent the whole work done by the driving
proeeures at eacli revolution of the imaginary aim, then

13 the projection of the space

./Google

THE CilANK GTUDB. 3B1

described by the extremity of the aiin during the ascending
and descending sti'okes respectively, therefore 2P, = — ^.
Substituting this value for 2P„

,1. j l-i^=li Bin. ».~^(?t rin. ,,+ ?-■ eia. ,.) \ =

U,+W*p,sin.(p, (332),

which is the modulus of the double crank, the directions of
the driving pressure and the resistance being both aupposed
vertical; or if the friction resulting from the weight of the
crank be neglected, and W be therefore assumed =0, then
does the above equation represent the modulus of the
double crank, whatever may be the direction of the driving
pressure, provided that the direction of the resistance be
parallel to it. Dividing by the coefficient of U„ and
neglecting terms of more than one dimension in sin. ?, and

— sin. 9, ) f + ■W*p,ein.9, .... (333).

The Crank GmoE.

264. In some of the most important applications of the
steam engine, the crank is made to receive its continuous
rotatory motion, from the alternating rectilinear motion of
the piston rod, directly through the connecting rod of the
crank, without the intei-vention of the beam or parallel
motion; the connecting rod being in this case jointed at one
extremity, to the extremity of tlie piston rod, and the obliqne
pressure upon it which resnlte from this connexion being
sustained by the intervention of a cross piece fixed upon it,
and moving between lateral guides.*

• This cortriTOnee is that well known as applied to the loeomotlve carriage.

, Google

THE CKANK GUIDE.

Let the length CD of the connecting rod be represented
by J>, and that BD of the cranli arm by a, and let P, and P,
in the above figure be taken respectively, to represent the
pressure npon iJie piston rod of the engine and the connect-
ing rod of tlie crank, and PS to represent the direction of
the resietance of the guide in tlie state bordering npon
motion by the excess of the driving pressure P,. Then is
KS inclined to a perpendicular to the direction of tiie guides,
or of the motion of the piston rod, at an angle equal to the
limiting angle of resistance (Art. 141) of the surfaces of con-
tact of the guides.

Since, moreover, P„ P,, E are pressures in equilibrium,

P, gin.F,CS
■■■ P, " sin. P,CS'

Let ZPCD=S ; limiting angle of resistance of guide =<p;
therefore, P,CS=^-p, p,C8=^-Hp-fl ;

-C-f)

Let BD=:a, CD = 5, and DEC = fl„ and assume P, to
remain constant, P, being made to vary according to the
conditions of the state bordering upon motion ;

.-. aU,=:P, . aAC=_P, . aBC=-P, . A(a COS. i,-Vl cos. i)~

P, sec. a? COS. (A— f) {ffisin. fl,AS,-f& sin. flA^) ;

Air^=-P,(ABC)cos.a=P,(«Bin.fl,A^,+5sin.Mfl)cos.fl;

;.ir,=P,8ee.'pl«/3in.(),cos.(fl™9)(7S, + 5 /sin.^co8.(C— 9)(?Sj.

./Google

D",=F, {afsm. i, cos. Sdi,+bfsm. S cos. S(M)

The second integral in each of these equations vanishes
between the prescrihed limits ; also sin. ^ = r sin. ^ ; there-
fore COS. () = (1— ^sin. °i,)';

.-. V,^F,afem. 6, cos. !>d^,^V,af{l - j, sin. '6,f sin. S^d6,=
—p^^yj |i-lj+cos.'''?Tc?cos.fl,=

U,=P,a sec. tp Ain. 6^ cos. {3— ?)(^,=P,«y sin. \ cos. fl(?fl,+

T,aiim. py sin. d sin. fl,i^a^="U",+P,^tan. ?- Ain. 'A, (2^,=

U, + P,|^tan.9;
whence eliminating P, and reducing, we ohtain

tl.=uj 1 +, .. '*""■' , 1 (338),

which is the modulus of the crank guide.

The Flt-Wheel.

265. The angular velocity of the fiy-wheel.

Let P, be taken to represent a constant pressure applied
through the connecting rod to the arm of the crank of a

• Cliurch'a Diff. and Int. Cal. Art. 199.

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THE FLT-WHEEL.

Bteam engine ; suppose tlie direction of this pressiu-e to
remain always parallel to itself, and let P, represent a con-
stant reeietance opposed to the revolntion of the axis which
caiTies the fly-wheel, by the nsefal ^ork done and the pre-
mdicjal resistances intei^osed between the axis of the
tfly-wheel and the working points of the machine.

Let the angle ACB=fl, CB=a, CP,=ffl,.

Now the projection, upon the direction of P„ of tlie path
of its point of application B to the crank arm, whilst that
arm describes the angle ACB, is AM, thei'efore (Art. 52.),
the work done by Pi upon the crank, whilst this angle is
■described, is represented hy P, . AM, or by Pi a vera. i.
And whilst the crank arm revolves throngh the angle fl, the
resistance P, is orercorue through the arc of a circle sub-
ttended by the same angle 5, but whose radius is a„ or
through a space represented by a,i. So that, neglecting the
Action of the crank itself, the work expended upon the
Tesistances opposed to its motion is represented hy PjtJs,^, and
the excess of the work done upon it through tlie angle AOB
by the moving power, over that expended during me same
jperiod upon the resistances, is represented by

P,avers.i-P,V (336).

Kow SftP, represents the work done hy the mo^ng pressure
P, during each efi"ective stroke of the piston, and S^ta^P^ the
work expended upon the resistance during each revolution
of the fly-wheel; so that if m represent the number of
strokes made by the piston whilst the fly-wheel makes one

./Google

THE FI.T-WKEEL, 35o

revolution, and if tlie engine be conceived to have attained
its state of unifoiin or steady action (Art. 146.}, tlieu

:.a,'P,=~aP, (337).

Eliminating from equation (336) the value of a,P, deter-
mined by this equation, we obtain for the excess of the work
done by the power (whilst the angle i is described by the
crank arm), over that expended upon the resistance, the
expression

P,<.|Yers.«-!^| (338).

But this excess is equal to the whole work which has been
accumnlating in the different moving parts of the machine,
whilst the angle i is described by the arm of the crank (Art.
145). Now, let the whole of this work he conceived to have
been accumulated in the fly-wheel, that wheel being pro-
posed to be constructed of such dimensions as sufficiently to
equalise the motion, even if no work accumulated at the
same time in other portions of the machinery (see Art. 150.),
or if the weights of the other moving elemenls, or their
velocities, were comparatively so small as to cause the work
accumulated in them to be exceedingly small aa compared
with the work accumulated during the same period in the
fly-wheel. Wow, if I represent the moment ot inertia of the
fly-wheel, f- the weight of a cubic foot of its material, a, its
angular velocity when the crank arm was in the position
CA, and a its angular velocity when the crank arm has

passed into the position CB ; then will ^ — (a'— a,") represent

the work accumulated in it (Art. 75.) between these two
positions of the crank arm, so that

i^

f.1

vers. ^—

266. The posithns of greatest and Imst angular velocity of
the fly-wheel.

If we conceive the engine to have acquired its state of
steady or uniform motion, the aggregate work done by the

./Google

THE FLY-WHEEL.

power being equal to that expended upon the r
then -will tlie angular velocity of tlie fly-wheel return to the
same value whenever the wheel returns to the same position ;
BO that the value of a, in equation (SSS) is a constant, and
the value of « a function of 3 ; a aseumes, therefore, ite mini-
mum and maximum values with this function of ^, or it is a

i<0. But^=:Bin.a-

=0, when

. (340.)

IRow this equation is evidently eatieiied hy two values of
6, one of which is the supplement of the other, so that if i
represent the one, then will {^f— i) represent the other;
which two values of 9 give opposite signs to the value cos.
B of the second differential co-efficient of a', the one heing
positive or >0, and the latter negative or <0. The one
value corresponds, tlierefore, to a minimum and the other
to a maximum value of a. If, then, we take the angle ACB

in the preceding figure, sucli that its sine may equal ~

(equation 340), then will the position CB of tlie cranli arm
he that which corresponds to tJie ininiraiim angular velocity

./Google

af)7

of tlie fly-wheel ; and if we make the angle ACE equal to
the supplement of AOB, then is OE the position of the
crank ann, which corresponds to the maximum angular
velocity of the fly-wheel.

26 T. The greatest variation of the angular vdocift/ oft A-.

Let Kj be taken to represent the least angular velocity of
the fly-wheel, corresponding to the position OB of the crant
ami, and a, its greatest angular velocity, corresponding tc

the position CE ; then does 5- U'-^a.,') represent the work

-^?
accumulated in the fly-wheel between these positions, which
accumulated work ia equal to the excess of that done by the
power over that expended upon the resistances whilst the
crank arm revolves from the one position into the other,
and is therefore represented by the difference of the values
given to the formula (338) when the two values f—i\ and
1], determined by equation (340), are substituted in it for i,
Sow this difference is represented by the formula

p,»j™™. (»-,)-«■,. ,-t^i^ I ,

orbyP.a j 2 cos. 1— mil — ^1 \ ;

...».---...=Sp{.c...,-»(l-|')| (Ml);

in which equation -n is taken (equation 340) to represent

. . «i
that angle whose sine is — .

268. The dimensions of thefiy-wheel, such thai its c ^
velocity may at no period of a revolution deviate ieyond

Let ^ he taken to represent the mean number of revo-
lt
lutions made by the fly-wheel per minute; then will ^^

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60a TIIJ! PLY-WHEEL.

represent the mean number of revolutions or parts of a

revolution made by it per second, and i7r^2ff, or -^^, the

mean epace described por second by a point in the fly-wheel
whose distance from the centre is unity, or the mean angular
velocity of the fly-wheel. Kow, let the dimensions of the
fly- wheel be supposed to be such as are snfiicient to cause
its angular velocity to deviate at no period of its revolution

by more than -th from its mean value ; or such that the raax-
and that its mhiimmn value o.^ may equal -^tA 1 ) ; then

imum value cc, of its angular velocity may e

' 60 \

Substituting in equation (341),

I'K' 2P,o(sl„ /, 3M 1

w»=^r 1 ^ "'• "-"l 1 - TJ ] ■

Let H be taken to represent the horses' power of the
engine, ^timated at its driving point or piston j then will
33000H represent the number of units of work done per
minnte, upon the piston. But this number of units of work
is also represented by JNm . 2P,a ; since j^Nm is the number
of strokes made by the piston per minute, and 2P,ffl is the
work done on the piston per stroke,

.■.2P,a=66l300^.

Substituting this value for SPjO in the above equation, we
obtain, by reduction,

^ j 66000.30V 1 J „ 1-, 2^\ I H'^ ,„,„,

Let k be taken to represent the radius of gyration of the
wheel, and M its volume ; then (Art. 80.) MF=I, therefore
IJ.'K.h'=iJ-I. But fiM represents the weight of the wlieel
in Iba. ; let W represent its weigiit in tons ; therefore,
11.^1=2240^. Substituting this value, and solving in
respect to "W",

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THE FLT-WHEEL. 359

„ j eeooo.so'.j. 1 i / a^i i ii«

Substituting their vtilaes for * and g, and determining tlie
immei'ical value of the co-efficient,

W=8»l||cos.,-(l-5)|i| (Ma).

If the influeuce of the work accumulated in the arms of
the wheel he given in, for an increase of the ec[ualising

Eower heyond me prescribed limits, that accumulated in the
eavy rim or ring which forms its periphery being alone
faben into the account;* then (Art. 86.) M^=I=3*JcR
(E'+io*), where 6 repreaente the thickness, c the depth, and
E the mean radius of the rira. But by Guldinus's first
property (Art. 38.), 2'tJcK=M; therefore J'^tE'+ic").
Substituting in equation (343)

V=86491 1 1 c». ,- ( 1 - 5) } jp(|^ .... (344).

If the depth e of the rim be (as it usually is) small as
compared with the moan radius of the wheel, ^' may be
neglected as compared with E', the aboye equation then
becomes

W=86491 j^ cos. >,-(l-^j |~g^ (345);

by which equation the weight W in tons of a fly-wheel of a
given mean radius E is deterinined, so that being applied to
an engine of a given horse power H, making a given num-
ber oi revolutions per minute ^S, it shall cause the angular

velocity of that wheel not to vary by more than -th from its

mean value. It is to be observed that the weight of the
wheel varies inversely as the cube of the number of strokes
made by the engine per minute, so that an engine making
twice as many strokes as another of equal horse power,

" If the BBotion of each arm be auppoaed uniform and represented by k, and
the amie be » in number, it ia easily shown from Arts. '79., Ql., that the
momentum of LuerUii, of each arm about its estremity is yery nearly repre-
sented by ^^(R— ic)', ivhore o represents the depth of the rim; so that the

whole momentum of inertia of tl\e arms is represented by ^(t(K— iej'iiiihieh
expres^on must be added to the momentum of the rim to determine the whole
momentum I of the wheel. It appears, however, expedient to give the inerlia
of the arms to the equalising power of ttia Wheel.

, Google

360 THE FLY-WHEEL,

would receive an equal steadiness of motion from a fly.
wheel of one eighth the weight; the mean radii of the
wheels heing the same.

If, in equation (343), we substitute for I its value 2tScK',
or 29-KIl' (representing by K the section he of the rim), and
if we suppose the wheel to be formed of cast iron of mean
quality, the weight of each cubic foot of which may be
assumed to be 450 lb., we shall obtain by reduction

If=»2l|^0OB.,-(l-5)|Jj (346);

by which equation is determined the mean radius R of a fly-
wheel of cast iron of a given section K, which being applied
to an engine of given horse power H, making a given num-
ber of revolutions ^N per minute, shall cause its angular

velocity not to deviate more than — th from the mean ; or

conversely, the mean radius being given, the section E may
be determined according to these conditions.

269. In the above equations, m is taken to represent tlie
number of effective strokes made by the piston of the engine
whilst the fly-wheel makes one revolution, and ■n to represent

that angle whose sine is — .

Let now the axis of the fly-wheel be supposed to be a
continuation of the axis of the crank, so that both turn with
the same angular velocity, as is usually the case ; and let its
application to the single-acting engine, the double-acting
engine, and to the double crank engine, be considered sepa-
rately.

1. In the szTi^le-aciin^ engine, but one effective stroke of
the piston is made whilst the fly-wheel makes each revolution.

In this case, therefore, m=\, and sin. vi=_=0'31830!)8=:

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Substituting iu equations (345) and (346),

■ (3«);

by which equations are determined, according to the pro-
posed conditions, the weight W in tons of a fly-wheel for a
aingle-aotmg engine, its mean radius in feet K being given,
and its material being any whatever; and also its mean
radins R in feet, its section (in square feet) K being given,
and its material being cast iron ot mean quahty ; and lastly,
the section K of its rim in squai'e feet, its mean radius II
being given, and its material being, as before, cast ii-on.

2. In ike double-acting engine^ two effective stroiies are
mads by the piston, whilst the fly-wheel makes one

2
revolution. In this cases therefore, m — % and sin. ji— -=

0-636619= sin. 89" 32'; therefore, cos. >] = -7712549 - -

39° 33' / 2ii \

-^-gQ-=-21963; therefore 1-— =-56074;

{2 /-, ^M

= -21051.

Substituting in equations (345) and (346),

by which equations the weight of the fly-wheel in tons, the
mean radius in feet, and the section ot the rim in squai-e
feet are determined for the double-acting engine,

3. In ths engine working with two cylinders cmd a double
crank, it has been shown (Ai-t. 263.) that the conditions of
the working of the two arms of a double crank are precisely
the same sis though the aggregate pressure 2P, upon their
extremities, were applied to the axis of the crank by the
intervention of a single arm and a single connecting rodj

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THE FRICTION C

the length of this arm being represented by — instead of «,

and its direction equally dividing the inclination of the arnia
of the double crank to one another.

Now, equations (345) and (346) show the proper dinien
sions of the fly-wheel to be wholly independent of tlie
length of the crank arm ; whence it follows that the dimen-
sion of a fly-wheel applicable to the double as well as a
single crank, are determined by those equationB as applied
to the case of a double-acting engin^ the pressure upon
whose piston rod is represented by 2F,. But in assuming
^Ntti . 2Pj«=33000H, we have assumed the pressure upon
the piston rod to be represented by P, ; to correct this error,
and to adapt equations (345) and (346) to the case of a
double crank engine, we must therefore substitute -JH for H
in those equations. We shall thus obtain

W=9103.5i|,.

E=^-^ff, K=™.|i (e«,;

by which, equations the dimensions of a fly-wheel necessary
to give tlie required steadiness of motion to a double crank
engine are detennined under the proposed conditions.

The PkICTION of the FLY-WnEEL.

270. "W" representing the weight of the wheel and ? the
limiting angle of resistance between the surface of its axis
and that of its bearings, sin. tp will represent its coefBcient
of friction (Art. 138.), and ~W sin. if, the resistance opposed
to its revolution by friction at the surface of its axis. Now,
whilst the wheel makes one revolution, this resistance is
overcome through a space equal to tlie circumference of tlio
axis, and represented by 2*p, if p be taken to represent the
radius of the axis. The work expended upon the friction of
the axis, during each complete revolution of tlie wheel, is
therefore represented by 3*pW sin. p ; and if N represent
the number of strokes made by the engine per minute, and

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MODULTJS OF THE CliAITK ANJJ FLY-WHEEL. OOd

per mimite, then -will the number of units of work expended
per minute, upon tlie friction of the axis be represented bj
NTp"W sin. ip ; and the number of horses' power, or the frac-
tional part of a horse's power thus expended, by

^'^^^P'"'"-^ (360).

33000 ^ '

If in tliis eqxiation we substitute for W the weight in lbs.
of the fly-wheel necessary to establish a given degree of
Bteadmess in the engine, as determined by equations (SiT),
(348), and (349), we Siall obtain for the horse power lost by
fiiction of the fly-wheel, in the single-acting engine, the
d&uble-aotina engine, and the dmihle crank engine,
tively, the formulfe

Hwp sin. (p

3882.5l|^', 1941-aoS^^ (351)

The Modulus of the Ckank amd Ply-wheel,

271. If S, represent the space traversed by the piston of
the engine in any given time, and a the radius of the crank,
W the weight of the fly-wheel in lbs,, and p the radius of its

axis, then will Sos represent the length of each stroke, ^ the
immber of strokes made in that time, and 2*pW sin. $ . _ i
or 'sWSi- sin. 9 the work expended upon the friction of the

fly-wheel during that time, which expre^ion being added to
the equation (329) representing the work necessaiy to cause
the crank to yield a given amount of work U, to the ma-
chine driven by it (independently of the work expended on
the friction of the fty-wbeel), will give the whole amount of
work which must be done upon the combination of the crank
and fly-wheel, to cause this given amount of work to be
yielded by it, on the machine which the crank drives. Let
this amount of work be represented by TJ„ tlien in the case
in which tlie directions of the driving pressure and the re-
iistanee upon tlie crank are parallel (equation (329), and the

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THE aOVEBNOE.

fi-icfioii of the crane guide is neglected, we obtain for tlie
modulus of the crank and fly-wheel in the double-acting

■ p,)|u,.

?(352).

The Goveknok.

273. This instrument is represented in the figui'e, under
that form in which it is most commonly applied to the steam
. engine. BD and (& are rodsiointed

■ at A upon the vertical spindle AF,

and at D and E npon the rode DP
and EP, whicli last ai-e again jointed
at their extremities to a collar fitted
accurately to the surface of the spin-
dle and moveable upon it. At their
extremities B and 0, the rods DB
and EC carry two heavy balls, and
being swept round by the spiudle —
which receives a rapid rotation al-
ways proportional to the speed of the
niachme, whose motion the governor
is intended to regidate — these arms
by their own centrifugal force, and
that of the balls, are made to separate, and thereby to cause
the collar at P to descend upon the spindle, carrying with it,
by the intervention of the slioulder, the extremity of a lever,
■whose motion controls the access of the moving power to
the driving point of the machine, closing the tlirottle valve
and shutting off the steam from the steam engine, or closing
the sluice and thus diminishing the s\ipply of water to the
water-wheel. Let P be taken to represent the pressure of
the exti'emity of the lever upon the collar, Q the strain
thereby produced upon each of the rods DP and EP in the
direction of its length, "W tlie weight of each of the balls, w
the weight of each of the rods BD and CE, AE=a, AD=J,
DP=e, I'AB=fl, APD=Si. Now upon either of these rods
as BD, tlie following pressures ai-e applied : the weight of
the ball and the weigiit of the rod acting vertically, the
centrifugal force of tlie ball and the centrifugal force of the
rod acting horizontally, the strain Q of tlie rod DP, and
the resistance of the axis A. If a. represent t)ie angular

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THE GOVERNOR.

velocity of the spindlo, — a' . FB, or — a'asin. e, will repre-
sent the centrifugal force upon the ball (equation 102),
and — «°a' sin. ^ cos. i its moment about the point A : also

tlie centrifugal force of the rod BD produces the same eiFect
as though its weight were coUected in its centre of gravity
(Art. 134.), whoeo distance from A 19 represented by i{a~i),
BO that the centrifugal force of the rod is represented by

^—a-Xa—h) sin. i, and ita moment about the point A by

i— a°(»— J)' sin. 6 COS. i. On the whole, therefore, the sum of

the moments of the centrifugal forces of the rod and hall are

represented by — \'Wa'-i'^{a~-h)''\ sin. S coa. fl. Now if ,■*

' represent the weight of each unit in the length of the rod,
IP = p(a + &) ; therefore Wa' + iw{a - hj = W «' + ^i^{a' ~ V)
(a—h). Let this quantity be represented by Wiffl",

.■.W,=W+i^^(l-3(«-J)....(353};

then will ^W,(j°sin.fl cos.^ represent the sum of the momenta

of the centrifugal forces of the rod and ball about A. More-
over, the sum of the momenta of the weights of the rod and
ball, about the same point, is evidently represented by Wa
t.mj + wi(a-b) sin. e, or )ij \Wa+^i>.{a'-l^)\ smj; let thia
quantity be represented by "W^a sin. 6,

.•.W,=W+Wl-^) (354).

Also the moment of Q about A=Q . AH=Q5 ein. (H^).
Therefore, by the principle of the equality of moments, ob-
serving that the centrifugal force of the rod and ball tend to
communicate motion in an opposite direction from their
weights and the pressnre Q,

~~W,a' ein. S cos. d=Q5 sin. (^ +&,) +'W',<i sin. 6.

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-JO'S THE GOVEKNOE,

Now P is the resiiltant of the pressures Q acting i:. tlie
directions of the rods PD and PE, and inclinef to one
anoUier at the angle ^\ ; therefore (equation 13),

P=2QcosJ,;
.-. Q sin. (fl + fl,) = ^P sm.(fl + fl,)^^p ^^.^^ ^ ^ ^^^_ ^ ^^ ^^^ _

But Bince the sides h and c of the triangle APD are oppo-
,gles

Bite to the angles ^i and 6, therefore 5Hi:_i__ ; therefore
sin. 6 0

3S. 1= l-ilsin.'

.'. Q sin. {S +a,) = -^P I sin. 6 + - sin. i cos. S f 1 — -Y^m.'i j [ .

Suhstitnting this value in the preceding equation, dividing
hy sin, 6, and writing (1— cos, 'i) for sin. 'A, we obtain

^cos.'d)~* I +W,a.. .(355);

which eqtiatioii, of four dimensions in terms of cos. B, being
solved in respect to that variable, determines the inclination
of the arms under a given angular velocity of the spindle.
It is, however, more commonly the case that the inclination
of the arms is eiren, and that the lengths of the arms,
or the weights ot the balls, are required to be determined,
so that this inclination may, under the proposed conditions,
be attained. In this case the values of W, and W, must be
substituted in the above equation irom equations (353) and
(354), and that equation solved in respect to a or "W".

The values of o and o ai-e determined by the position on
the spindle, to which it is proposed to make tlie collar
d^cend, at the given inclination of the arms or value of i.
K the distance AP, of tJiis position of the collar from A, be
represented by ^, we have A=Jcos,i + c cos.*,,

, . (356) \

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THF. GOVEBNOE.

3(57

from wHch eqiiatioii and the preceding, tlie value of o^e
of the c[naiitities b or c may be aetermined, according to tiie
proposed conditions, the value of the other being assumed to
be any whatever.

If fT represent the number of revolntions, or parts of a
revolution, made per second by the fly-wheel, and yS the
number of revolutions made in the same time fay the spindle
of the governor, then will 2*7N represent the space a, de-
scribed per second by a point, situated at distance unity from
the axis of the spindle. Substituting this value for a in
eq^nation (355), ana assuming 5:=c, we obtain

^--W,a'cOB.9=P&-l-W,ffl . . . . (357):

also by equation (356),

/(=2icos.fl (358).

Eliminating cos. fl between these equations, and solving in
respect to X,

Let P (1+^) and P (1—^) represent the values of P
corresponding to the two states bordering upon motion
(Art. 140) and let N (I + 5) and N (1—^ be the correspond-
ing values of N ; so that the variation either way of -th from
the mean number N of revolutions, may be upon the point
of causing the valve to move. If these values be respectively
substituted for P and 1:1 in the above formula, it is evident
that the corresponding values of A wiU be equal. Equating
those values of A and reducing, we obtain

Bj which equation there is established that relation between
the quantities "Wa, a, F, m which must obtain, in order that a
variation of the number of revolutions, ever so little greater

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! CAEEL4GB-WHKEL.

than tlie -th part, may cauee the valve to move. Keglect
ing - as small when compared with n.

■ \' ' Pi)'

■which expression, representing that fractional variation in the
number of revolutions which is sufficient to give motion to
tlio valve, is the tnie measure of the SENsmiLiTT of the
governor.

373. The joints E and D are sometimes
fixed upon the arms AB and AO as in the
accompanying figure, instead of upon the
prolongations of those arms as in the pre-
ceding figure. All the fonnulfe of the
last Article evidently adapt themselves
to this case, if b be assumed =0 (in equa-
tions 353, 354). The centi'ifiigal force of
the rods EP and DP is neglected in this
computation.

The Cakeiage-wheel.

274. "Wliatever be the nature of the resistance opposed to
the motion of a carriage-wheel, it is evidently equivalent to
that of an obstacle, real or imaginary, which the wheel may
be supposed, at every instant, to be in the act of eurmount-
ing. Lideed it is certain, that, however yielding may he the
tnaterial of the road, yet by reason of its compression before
the wheel, such an immoveahle obstacle, of exceedingly small
height, is continually in the act of heing presented to it.

275, The two-wheeled carnage.

Let AB represent one of the wheels of a two-wheeled
carriage, EF an inchned plane, which it is in the act of as-
cending, O a solid elevation of the surface of the plane, or an
obstacle which it is at any instant in the act of passing over,

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THE CAKEI AGE-WHEEL.

P the corresponding trac-
tion, W the weight of the
wlicel and of the load whicK

Now the surface of the iox
of the wheel being in the
state bordering upon motion
. on the surface ot the axle,
the direction of the resist-
ance of the one upon the
other is inclined at the limit-
ing angle of resistance, to a
radius of the axle at their
point of contact {Art. 141.).
This resistance has, more-
over, its direction through
the point of contact O of
the tire of the wheel with the obstacle on which it is in the
act of turning. If, therefore, OK be drawn intersecting the
circumference of the axis in a point o, such that the angle
CoR may equal the limiting angle of resistance 9, then will
its direction be tJiat of the resistance of the obBtacle upon
the wheel.

Draw the -pertical GH representing the weight "W, and
through H draw HK parallel to OR, then will this line
represent (to the same scale) the resistance K, and GK the'
traction P (Art. 14.) ;

P_ GK sin. GHK sin. GHK

■■ W^GH-Bin. GKH-sin. (PGH-GHE:)=
sin. WsO

n. {PLW-W«0)'

Let E=radiua of wheel, p=radius of axle, AC0=:J(, AOW
=i=inclination of the road to the horizon, fl=inclination of
direction of the traction to the road. Now 'WsO='W0O-|-

COs, but WCO=i+)j, and !HL^^.^. Let COs bo re-
" sin. GoK CO

presented by «, then 'WsO=i+ij+a, and

AIsoPL"W"=^-hi-|-«; therefore PL'W"-"W5(

-(.,+a-fl);

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.,P=wSii±ltil (361);

COB. {^+a—0) ^ ' '

when the direction of traction is pai-allel to the road, 6=0,
:.l*=W\sm. t+ COS. I tan. (i+a)i .... (363).
I£ the road and the direction of traction he both horizontal
a=:(=0, and

P="W" tan. (>]+«) (363).

In all cases of traction with wheels of the common dimen-
sions upon ordinary roads, AGO or ■>] is an exceedingly email
angle ; a is also, m all cases, an exceedingly sm^l angle
(ecLuation 360); therefore tan. (?;+ffi)=?j + a yery nearly.
Sow if A be taken to represent the arc AO, whose length
is determined hy the height of the obstacle and the radius
of the wheel, then

A

"R. '

. (364).

SubetitTiting the value of k from equation (360),

p^^V^^p^ (365).

276. It remains to determine the value of the arc A inter-
cepted between the lowest point to which the wheel sinks in
ilie road, and the summit O of the obstacle, which it is; at
every instant surmounting. Now, the experiments of Cou-
lomb, and the more recent experiments of M. Morin,* ap-
pear to have fully established the fact, that, on horizontal
roads of uniform quality and material, tJie traction P, when
its direction is horizontal, varies directly as the load W, and
inversely as the radius R of the wheel; whence it follows
(equation 365), that the arc A is constant, or that it is the
same for the same quality of road, wiiatever may be tJie
weight of the load, or the dimensions of the wheel.f The

" Espfoiences aur Ic Tirage des Voitares, faitea en 1831 et 1838. (See Af-

\ In explanation of this fact let it be obaerved, that althoitgli the -whepl
sinks deeper beneath the surlaoe of the road ns the material is softer, yet the
obfltAcle yields, for the same leason, more. under the presanre of the wheal, the
arc A being by the one canse IncreBaed, and by the other diminished. Also,
that although by increaang the diameter of the wheel the arc A would he ren-
dered greater if the wheel sank to the aame depth as before, yet that it does
not sink to the same depth by reason of the corresponding increase of the sur-
iace which austaias the preseuro.

, Google

S CAKKIiGE-WITEEL.

coTiBtant A may therefore be taken as a measure of tlie re-
sisting quality of the road, and may be called the modul/us
of its redet(moe.

The mean value of this modulus being determined in re-
spect to a road, whoso surface ie of any given quality, the
ralue of tj will be known from equation (364), and the rela-
tion between the traction and the load upon that road, under
all circumstances ; it being observed, that, since the arc A
is the same on a horizontal road, whatever be the load, if tiie
traction be parallel, it is also the same under the same cir-
cumstances upon a sloping road ; the effect of the slope be-
ing equivalent to a variation of the load. The same substi-
tution may therefore be made for tan, {^ + 0.) in equation
(362), 38 was made in equation (363),

277. The hest direction, of traction m the two-wheeled

This beet direction of traction is evidently that which gives
to the denominator of equation (361) its greatest value ; it
is therefore determined by the equation

278. The fom^-wheeled

Let W„ W, represent the loads borne by the fore and
hind wheels, together with their own weights, Ei, E, their
radii, p„ p, the radii of their axles, and ip,, ip, the limiting an-
gles of resistance. Suppose the direction of the traction F
parallel to the road, then, since this traction equals the sums
of the tractions upon the fore and hind wheels respectively,
we have by equation (366)

P=W, I .in. ,+<A±fL*lli)c„,. , I +

wJ™.,+(A±P|*^)c.

, Google

THE CAEKIAGE-WHEBL.

279. The, work acoumulated in the carriage-wTieel*

Let I represent the moment of inertia of the wheel about
ita axis and M its volume ; then will MR'+I represent ita
moment of inertia (Art. 79.) abont the point in its circum-
ferences about which it is, at every instant of its motion, in
the act of turning. If, therefore, a represent its angular
velocity about this point at any instant, U the work at that
instant accumulated in it, and ji. the weight of each cnbical

unit of its mass, then (Art. 75.), XJ"=V-{MR'' + ^=:i-K

{all)'+ja'-l. Now if V represent the velocity of the axis
ofthe wheel, «E^Y;

whence it follows, that the whole work accumulated in the
rolling wheel is equal to the sum obtained by adding the
work which would have been accumulated in it if it had
moved with its motion of translation only, to that which
would have been accumulated in it if it had moved with its
motion of rotation only. If we represent the radius of gyra-
tion ("Art. 80.) by X, I^MK"; whence substituting and
reducmg,

u=j^M(i + g)T- {my

The accumulated work is therefoi-e the same as though the
wheel had moved with a motion of translation only, but with

a greater velocity, represented by the expression ( 1 + -0-5 ) V

" For a further discussion of the conditions of the rolling of a wlieel, see a
paper in the AppendlK on the Rolling Motion of a G;;linder.

I The angulai' Telocity of the wheel would evidently be a, If its centre were
fised, and ita circumference made to revolve with tlie same velocity as now.

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ACCELEBAXED OE MSTAEDED MOTIOH.

280. On the state of the ACOEtEKATEO OE THE BETAKDED
MOTION OE A MACHINE.

Let the work U, done upon the driving point of a machine
be conceived to be in excess of that U, yielded upon tlie
working points of the machine and that expended upon its
prejudicial resistances. Then we have by equation (117)

U,=AU,+BS,+~(V=-V,')2«J>.';

where T represents the velocity of the driving point of the
machine after the work XJi has been done upon it, Y, that
when it began to be done, and ^wX" the coefficient of equable
motion. Now let S, represent the space through which TJ,
18 done, and 8, Uiat through which U, is done ; and let the
above equation be differentiated in respect to Sj,

^_ (TO,

tnt the driving pressure.
A1bo"3c^ = P„ if Pj represent the working pressure; also

_dy '<pr ,it ^^ dv 1 dv „ . ^. ^,^

If, therefore, we represent hyAtherelation-jo^, between the
spaces described in the same exceedingly small time by the
driving and working points, we have

P,=AAP, + B+^s«»V (370);

P^_AAI^-B

wherey (Art. 95.) represents the additional velocity actually
acquired per second by the diiving point of the machine, if
P, and P, be constant quantities, or, if not, the additional
velocity which would be acquired in any given second, if
these pressures retained, throughout that second, the Taluea
which they had at its commencement.

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THE AOCELEEATION (

281, To deiermins the G06;ffiaiemt of equable motion.

SmjX' represents the sum of tlie ■weights of all the moving
elements of the machine, each being multiplied by the ratio
of ite velocity to that of liie driving point, which sum has
been called (Art. 151.) the coeffiderd of eqwaUe motion. K
the motion of each element of the macnine takes place about
a fixed axis, and a„ ti^, a,-, &c., represent the perpendictalara
from their several axes upon the directions in wliich they
receive the driving pressui-ea of the elements which precede
them in the series, and b„ J„ i„ &e., the similar perpen-
diculars upon the tangents to their common surfaces at the
points vehere they drive those that follow them; tiien,
while the first driving point describes tlie small space aS,,
the point of contact of the ^th and »+lth elements of the
series will be made (Art. 234.) to describe a space repre-
sented by

BO that the angular velocity of the j^th element will be
represented by

i.t. . . . t^i^g

a,a, . . .Op "

and the space described by a particle situated at distance f
from the axis of that element by

and the ratio >- of tliis space to that described by the driving
point of tlie machine will be represented by

The sum 2wX' will therefore be represented in
this one element by

\a,.a^ . . . ap I '^

Or if Ip represent the moment of inertia of the element, and
|j-B the weight of eacli cubic unit of its mass, that portion of
the value of 2wX" which depends upon this element will be
repreiiented by

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OF TOB MOTION Of A MACHINE. 375

/5,5, . . . J>^-A\

\ a,a^ . . . Op I
And the same being tme of every other element of the
machine, we have

which is a general expression for the coefdcient of equable
motion ia the case supposed. The value of A in equation
(STl) is evidently represented by

^82. To determine thepressure upon thepomt of contact of
a/ny inm elements of a machine moving with am. aooelerated
or retarded motion.

Let^, be taken to represent the resistance upon the point
of contact of the firet element with the second, _p, that upon
the point of contact of the second element of the machine
with the third, and so on. Then by equation (370), obeerv-
ing that, P, and jj, representing pressures applied to the
same element, ^w>^ is to be taken in this case only in
respect to that element, so that it ia represented by fj-,!,,

whilst A ia in this case represented by — ', we have, neglect-
ing friction,

Substituting the value of /"from equation (371), and solving
in respect to_^„

^.=ti'.~l;(j'.-^-)ss. («*).

where the value of A is determined by equation (373), and
that of 2w^.° by equation (37^). Proceeding similarly in
respect to the second element, and observing that the
impressed pressures upon that element are p, and j>„ we
have

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67Q ACCELERATED OB EBTAEDED MOTION.

y, representing the additional velocity per second of tho

point of application of jj„ which evidently equals — /.

Suhetituting, therefore, the value of/ from equation (3T1)
as before,

Substituting the value of p^ from equation {37i), and solv-
ing in respect to^p,, we have

!'-=if^'^ifi. \ '-•^ + \«J '■•^- \ (ts^) • ■ • • (''">■

And proceeding similarly in respect to tlie other points of
contact, the pressure upon each may be determined. It ia
evident, that by assuming values of A and B in equations
(370) and (371) to represent the coefficients of the moduh in
respect to the several elements of the machine, and to the
whole machine, tlie influence of friction might, by similar
steps, have been included in the result.

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THEOET or THE STABILITY OF STEUCT0EES.

Geneeai CoNJjrrioNs of the Stabilitt of a S'
Uncemktisip Stones.*

A BTRirOTtrEE may yield, under tlie pressures to which it is
subjected, either by the slipping of certaio of its surfaces of
contact upon one another, or by their turning over upon the
edges of one another ; and these two conditions involve the
■whole question of its stability.

The Lixe of Hesistaxce.

283. Let a structure MNLK, composed of a single row of
nncomented stones of any forms,
, and placed under any givcQ circum-
stances of pressure, be conceived to
i mtersected by any geometrical
mrface 1 2, and let the resultant a A
f all the pressures which act upon
one of the parts MNSl, into which
this intersecting surface divides tlie
strocture, he imagined to be taken.
Conceive, then, this intersecting
surface to change its form and posi-
tion so as to coincide in succession
with all the common surfaces of
contact S 4, 5 6, 7 8, 9 10, of the
atones which compose the stmctm'e :
and let 5B, cG, dD, eE be the re-

* Extracted from d memoir on tbe Theoir of tlie Arch by the author of this
work in the first volnme of the " Theoretical and Practical Treatise on Biidges,"
by Professor Hosting luid Mr. Hann of King's College, published by Mr. Wealo.
These general conditions of the equilibrium of a system of bodies in contact
were first diecuBsed by the author in. the fifth and sistb volumes of the " Carar
bridge PhiloEophieal TranBaotlons."

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378 THE LDiE OF EESI3TANCE.

sultants, similarly taken with tsA, which correapoiid to those
several planes of intiirsection.

Ill each such position of the mtersecting surface, the restilt-
aut spoten of having its direction prodiiced, ■will intersect
that surface either within the mass of tlie structure, or, when
that surface is imagined to be produced, without it. If it
intersect it without the mass of the structure, then the whole
pressure upon one of the parts, acting iu the direction of
this resultant, will cause tliat part to turn over upon the
edge of its common surface of contact with the other part ;
if it intersect it withm the mass of the structure, it will not.

Thus, for instance, if the direction of tlie resultant of the
forces acting upon the part NM 1 2 had been a' A', not inter-
secting the surface of contact 1 2 within the muss of the
structure, but when imagined to be produced beyond it to »' ;
then the whole pressure upon this part acting in a' A! would
have caused it to turn upon the edge 3 of the sni-face of con-
tact 1 2 ; and similarly if the resultant had been in a" A",
then it would have caused the mass to revolve upon the
edge 1. The resultant having the direction aA, the mass
■will not be made to revolve on either edge of the surface of
contact 1 2.

Thus the condition that no two parts of the mass should be
made, by the insistent pressui'es, to turn over upon the edge
of their common surface of contact, is involved in this other,
that the direction of the resultant, taken in respect to every
position of the intersecting surfece, shall intersect that sur-
iace actually within the mass of the structure.

If the intersecting surface be imagined to take up an *»hJ-
nite number of different positions, 1 2, 3 4, 5 6, &c., and the
intersections with it, a, h, o, d, &c., of the directions of all
the corresponding resultants be found, then the curved line
(Aodef, joining mese points of intersection, may with pro-
priety be called the lihe as eesistance, the resisting points
of the resultant pressures upon the contiguous surfaces lying
all in that line.

This line can be completely detei-mined by the methods of
analysis, in respect to a structure of any given geometrical
form, having its parts in contact by suiiaces also of given

feometrical fonne. And, conveisely, the form of this line
eing assumed, and the direction which it shall have through
any proposed structure, the geometrical form of that struc-
ture may be determined, subject to these conditions; or
lastly, certain conditions being assumed, both as it regards
the form of the structure and its hne of resistance, all that is

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THE LEJE OF PEEaSBEB. 379

ssary to the existence of these assumed conditions may
be found. Let the stracture ABCD have for its line of re-
sistance the line PQ. Now
-~f it is dear t!iat if tliis line
cut the suiface MN of any
section of the mass in a point
n without the surface of the
ma^, then the resultant of
the pre^ures upon the mass
CMS will act through n,
and cause this portion of the
mass to revolve about the
nearest point N of the in-
tcTsection of the surface of
section MN with the surface of the strncture.

Thus, then, it is a condition of the ecLaiUbrinm that the
Ime of resistance shaU mterseci the common- swface of oon-
taot of each two contiguous portions of the structwe actuaWy
toitkm the mass of the stritotwe ; or, in other words, that it
shall actually go through each joint of the structure, avoid-
ing none : this condition being necessary, that no two por-
tions of the structure may revolve on the edges of their
common surface of contact.

The Iinb op Peessurb.

284. But besides the condition that no two parts of the
structai'c should turn upon tlie edges of their common sur-
faces of contact, which condition is involved in the determi-
nation of the LINE OF BESiSTAHoffi, there is a secoad condition
necessary to the stability of the structure. Its surfaces of
contact must no where slip upon One another. That this
condition may obtain, the resultant corresponding to each
surface of contact must have its direction withm certain
limits. Tliose limits are defined by tlie eurface of a right
cone (Art. 139.), having the normal to the common surface
of contact ,at the above-mentioned point of intersection of
the resultant) for its axis, and having for its vertical angle
twice that whose tangent is tlie co-efffcient of friction of the
surfaces. If tlie direction of the re8ult,ant be within this
cone, the surfaces of contact will not slip upon one another ;
if it be without it, they v^ill.

Thus, then, the directions of the consecutive resultants in

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380

[Lrrr of a solid i

respect to the normal to the point, where each intersects its
corresponding surface of contact, are to be considered as im-
portant elements of the theory.

Lot then a line ABODE be taken, which is the locus of
the consecutive intei-sections of tlie
resultants «A, iB, oO, dl>, &c. The
direction of the resultant pressure
upon eveiy section is" a ta/ngmA to
this line ; it may therefore with pro-
priety be called the line of peessuke.
Its geometrical form may be deter-
mined under the same circumstances
as that of the line of resistance. A
straight line oC drawn from the point
(3, where the line of eesistance ahed
intersects any joint 5 6 of the struc-
ture, so as to touch the uhe git pebs-
suEE ABCD, will detenniue the
direction of the resultant pressure
upon that ioint; if it lie within the cone spoken of, the
structure will not slip upon that joint ; if it lie without it,
it will.

Thus the whole theory of the equilibrium of any structure
is involved in the detennination with respect to that struc-
tm'e of these two lines— -the line of resistance, and the line
of pressure : o«e of these lines, the line of resistance, de-
termining the point of application of the resultant of the
pressures upon each of the surfaces of contact of the system ;
and the other, the line of pressure, the dwection. of that
resultant.

Tile determination of both, under their most general forms,
lies within the resources of analysis ; and general equations
for their determination in that case, in which all the surfaces
of contact, or joints, are planes — the only case wliieh offers
itself as ajiracticcd case— have been given by the author of
this work in the sixth volume of the " Cambridge Philo-
sophical Transactions."

The Stabilut of a Solid Body.

285. The stability of a solid body may be considered to be

freater or less, as a greater or less amount of work must be
one upon it to overthrow it; or according as the amount

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THE STABILITT OF A aTRCCTUItE. 3S1

of ivork which mu&t be doiie upou it to 'hriii^ it into
that position in -which it will fall over of its owii accord is
greater or leas. Thus the stability of the solid represented
m Jig. 1. resting on a horizontal p,^^ ^ ^

plane is greater or less, according

as the work which must be done i — ■ — n /"^""-^^

upon it, to bring it into the position I / J

represented in^g. 2,, where its cen- / • /

tre of gravity is in the vertical \ / /j /

passing through its point of su^ \ / /' j /

port, is greater or less. Now this ; ''-J ^^-^j/

wor> is equal (Art. 60.) to that

wtiieh would be necessary to raise its whole weight, verti-
cally, through that height by which its centre of gravity
is raised, in passing from the one position into the other.
Whence it follows that the stabiliLy of a solid body resting
npon a plane is greater or less, as the product of its weight
by the vertical height throogh which its centre of gravity k
raised, when the body is brought into a position in whicn it
will fall over of its own accord, is greatei' or leas.

It' the base of the body be a ^ane, and if the vertical
height of its centre of gravity when it rests npon a horizontal
plane be represented by A, and the distance of the point or
the edge, upon which it is to be overthrown, from the point
where its base is intereected by the vertical through its
centre of gravity, by h ; then is the height through wliicli its
centre of gravity is raised, when the body is brought into a
position in which it will fall over, evidently represented by
(A' + F)*— A; so that if W represent its weighty and IT the
work necessary to overthrow it, then

U=W )(A'+;t=)^-A5 .... (STfl).

D is a true measure of the stability of the body.

Thi! SiABH^rrT OF A Structitee.

386. It is evident that the degree of the eta.bility of a
structure, composed of any number of separate but contigu-
ous solid bodies, depends upon the less or greater degree of
approach which the line of resistance maltes to the extradoa
or extei-nal face of the stnicture; for the structure cannot be
thrown over until the line of resistance is so defected as to

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3S2 Till.; WALL OK HKR.

interaect the extrados : the more remote is its direction from
tliat suj-'face, when free from any exti-aordinarj pressure, the
leee is therefore the probability that any snch pressure will
overthrow it. The ueai-est distance to which the line of re-
sistance approaches the extrados will, in the following pages,
he represented by m, and will be called the Monuioa of
SxABiLrry of the structure.

This shortest distance presents itself in the wall and but-
tress commonly at the lowest section of the structure. It is
evidently beneath that point where the line of resistance in-
teraects the lowest section of the structure that the greatest
resistance of the foundation should be opposed. If that point
be iirmly supported, no settlement of the stracture can take
place under the influence of the pressures to which it is ordi-
narily subjected,*

The "Wall oe Piek.
287. The staMUty of a wall.

If the pressure upon a wall he uniformly distributed along
its length,! and if we conceive it to be intersected by verti-
cal planes, equidistant from one another aud perpendicular
to its face, dividing it into separate portions, then are the
conditions of its st^ility, in respect to the pressures applied
to its entire length, manifestly the same with the conditions
the stability of each of the individual portions into which it
is thus divided, in respect to the pressures sustained by that
portion of the wall ; so that if every such columnar portion
or pier into which the wall is thus divided be constracted so
as to stand under its insistent pressures with any degree of
firmness or stability,, then will the whole structure stand with
the like degree of firmness or stability ; and convereely.

In the following discussion these ec[ual divisions of the
length of a wall or pier will be conceived to be made one
toot apart ; so that in every case the question investigated
will be that of the stability of a column of unifoi-m or varia-

' A practical rule of Tauban, genenUlj adopted in fordfications, brings the
point where the line of resistance intersects the base of tlie wall, to a dietacoe
from the lerlioal to ita centre of gravity, of f Ihs the distance from the latter
to the external edge of tlie base. (See Poncelct, MSewtre mr la Stabiliti ilea
Jleiielem«ii9, note, p. S.)

f In the wall of a building the pressure of the rtiftere of the ro6f is tliM
anifornily distributed by the interyeutiou of the ivall pktcs.

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C LINB OF KESISTAN'CK IN j

ble thickness, whose width meaa^ired in the direction of the
length of tlie wall ie one foot.

288. When a wall is supported by buttresses placed at
eqoal distances apart, the conditions of the stability will be
made to resolve themselves into those of a continuous wall,
if we conceive each buttress to be ex-
tended laterally until it meets the adja^
cent huttress, its material at the same
time so diminishing its specific gravity
that its weight when thus spread along
the face of the wall may remain the
same as before. There will thus be ob-
tained a compoaud wait whose external
and internal portions are of different
speciiic gravities ; the conditions of
whose equilibrium remain manifestly
unchanged hy the hypothesis which has
been made in respect to it.

The Line of Resistance ts a Pier.

28&. Let ABEF be taken to repre-
sent a column of uniform dimensions.
Let PS be the direction of any pres-
yf sure P sustained by it, intersecting its
J, a/' axis in O. Draw any horizontal sec-
tion IK, and take ON to represent
the weight of the portion AKIB of
the column, and OS on the same scale
to represent the pressure P, and com-
plete the parallelogram ONES ; then
will OR evidently represent, in mag-
nitude and direction, the I'Csultant of
the pi-essurea upon the portion AKIB
of the maea (Art. 3.), and its point of
intersection Q with IK will represent
a point in the line of resistance.
Let PS intersect BA produced if necessary) in G, and let
G0=&, AB=», AK=iC, MQ=w, POC=<x, (i-weight of
each cubic foot of the material of the mass. Draw KL per-
pendicular to CD; then, by similar triangles.

vR"

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SSi TEIE LINS OF EESISTANCE ES A PIEE.

QM_RL

OM~OL

But QM^y, OM^CM-CO^^c-Z; .cot._^ EL=KK
sin. RNL:^P sin. a, OL=ON+KL=ON+KK cos. EKL
=)!:«!(; 4-P COS. a ;

—k cot. a fi(IiC + P COS.
n. ct— ^ COS. a

(3n) ;

which is the general ecuiation of the line of I'esistance of a
pier or wall.

290. The conditions necessary that the stones of the pier may
not slip on one another.

Since in the construction of the parallelogi'tLm ONKS,
whose diagonal OK detennines the direction ot the resultant
pressure Tipon any section IK, the side OS, representing the
pressure P in magnitude and direction, remains always the
same, ■whatever may be the position of IK ; whilst tlio side
OS, repreBenting the weight of AKIB, increases as IK de-
scends: the angle KOM continually diminish^ as IK de-
scends. Now, this angle is evidently equal to that made by
OR with the perpendicular to IK at Q ; if, therefore, this
angle be less than the limiting angle of resistance in the
highest position of TK, then ■will it he less in every sahjacent
position. But in the highest position of IK, ON=0, so that
m this position K.OM=a. No^w, so long as the inclination
of OR to the perpendicular to IK is less tlian the limiting
angle of resistance, the two portions of the pier separated by
that section cannot slip upon one another (Art. 141.). It is
therefore necessary, and sufficient to the condition that no
two parts of tlie structure should slip upon their common
surface of contact, that the inclination ci of P to the vertical
should be less than the hmiting angle of resistance of the
m surfaces of the stones. All the resultant pressurea
5 through the point 0, it is evident that the Une qf
e (Art. 384.) resolves itself into timtpoinf.

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THE LINE OF

291, The greatest height of ths pi,er.

At the point where tlie line of resistance intersects the
external face or extrados of Hie pier, y=^a ; if, therefore, H
represent* tlie corresponding value of x, it will manifestly
represent the greatest height to which the pier can be built,
80 as to stand under the given ii^iatent pressure P. Substi-
tuting these values for x and y in equation (STT), and solving
ju respect to H,

jj^rvs»+«;j^ ^3^gj^

_P(iix+ifc)cos.ix
a— ^'

K P sin, a=-|fKi.°, 'S.=infim^ : whence it follows that in
this case the pier will stand under the given pressure P how-
ever great may be the height to which it is raised.

292. The Une of resistanse is an equilateral hyperbola.

Multiplying both sides of equation (377) by the don nni-
nator of the iraction in the second member,

^{(j-OiB-l-P COS. o,)=Pa! sin. ct— P^ cos, a;

dividing by i^a, transposing, and changing the signs of all the
terms,

Psin.a / PcOS. a\ P COS. a

lid " \ t>.a I '~ i>.a '

Psin

^(^+^

/Psm.a \/ Pcos.ai Pcos.a/, P sm. o
\ i^a " I \ ii-a I >^a \ t^a

T ^ fin 1 .1 1^ Psin. a ,,„ Pcos. d
Let Oil he taken ciua] to ■ — -. TIT=— : an

VQ=y„ TV=»„

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THE LINE OP RliSTSTiKCE IN A PIEK.

.■.y,= V"Q=VM-MQ=CH-MQ =

= a constant quantity.

B TX.* Ihe line of resist-
mtinually approaches TX
therefore, but never meets it ; whence
S- it follows, that if TX lie (as shown
-•^ in the figure) ivithin the surface of
the mass, or if C H < C B or
Psin a
<-ka, or 2P sin. tt<w.a', then

the line of resistance will no where
cut the extrados, and the stiiictui-e
will retain its stability under the in-
sistent pressure P, however high it
may be built ; which agrees with
the result obtained in the preceding
article.

ier, so that when raised to a given
height it may haoe a given stahiUty.

Let m be taken to represent the nearest distance to which
the line of resistance is mtended to approach the extrados of
the pier, which distance determines the degree of ite stability,
and has been called the modvhis of stability (Art. 286.). It
is evident irom tlie last article that this least distance will
present itself in the lowest section of the pier. At this
lowest. section, therefore, y^^a—m. Substituting this value
for y in equation (377), and also the height h of the pier for
X, and solving the resulting quadratic equation in respect to
a, >we shall thus obtain

/P cos. a. \
'^=-1-2-^1--^/ +

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L WALL SUPrOKTED BY S

294. To va/ry the point of wppUoation of the pressure P, so
that any required stability may be given to the pier.

It is evident, that if id equation (377) we substitute fa— wi
for y, and k for x, me modulus of stability m
may be made to assume any given value for
a given thickness a of the pier, by assigning
a corresponding value to h ; that is, by mov-
ing the point of application G to a certain
distance from the axis of the pier, deter-
mined by the value of k in that equation.
This may be done by various expedients,
and among others by that shown in the
figure. Solving equation (37T) in respect to
A, we have

v-ah

^^ MP cos. a)

It is necessary to the eciuilibrium of the pier, undei' these
circumstances, that the line of resistance should no where
intersect its intrados below the point D.

The STABn.nT of a "Wall supposted by Shoees.

195. Let the weight of the portion of the wall supported
by each shore or prop, and the
I pressure insistent upon it, be im-
agined to be collected in a singln
foot of the length of the wall ; th';
conditions of the stability of tho
wall evidently remain unchanged
by this hypothesis. Let ABOD
represent one of the columns or
piers into which the wall will thus
be divided, EF the corresponding
shore, P the pressure sustained upon
the summit of the wall, Q the
tbiTist upon the shore EP, 2w its
weight, X the point where the line
of resistance intersects the base of
the wall, Gx=m, CF^b, FEO=,S;
and let the same notation be taken in other respects as in

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dS9 A WALL S-CPPOETED BY SHOEEB.

the preceding articles. Then, since ic ia a point in the direc-
tion of the reeultant of. the resistances by which the base of
the column is sustained, the sum of the moments abont that
point of the pr^sure P and half the weight of the shore,
supposed to be placed at E*, is equal to the sum of the
momftnte of the tnnist Q, and the weight f-ah of the column ;
or drawing xM. and ieN perpendiculars upon the directions
of P and Q,

P. ^+w . 5C=Q .'x^+i'-ah . xK.

Now xM = '^s sin. i»sM=(HX— H*) sin. a = \h—(Rp+st)
cot. a} sin. a.=ABin. a~{h+ia—mj cos. a, xS =(p-\-m) cob. /3,

.'.PJAsin.a— (^+i«— «i.)cos.a{ +

wm=Q(J+m)cos. /3 + |j,fflA(^— m)

Solving thig equation in respect to Q, and reducing, we
obtain.

V \/i &m. a^Qc -\- ia)cos. a\-~it^a'h+m(P COB, a. + iJ'ah + v})
^ - (5 + m)eos./St ^TT^!

This expression may be placed under the form
Q=(Pcos. a+)^aA+-w)sec. /3—
Pjicoa. a— /tsii). a + (£+i(()coe. a} +fi«^{^a+5)+w5
' ■ (5 + «i)cos. /3f

If the numerator of the fraction in the second member of
this equation be a positive quantity {as in all practical cases
it will probably be found to be) the value of Q manifestly
diminishes with that of m. Now the least value of m, con-
sistent with the stability of the wall, is zero, since the hno
of resistance no where intersects the extnidos; the least
value of Q (the shore being supposed necessary to the sup-
port of the wall) corr^pon<£, therefore, to the value zero of
m ; moreover tliis least value of the thrust upon the shore
consistent with tlie stability of the wall is manifestly that
which it sustains when the wall simply rests upon it, the

• The weight 2to of che ehope may be ooiioeJTed to be divided into two aquiJ
parts and collected at its extremitioa.

f Tte expresdon {i+m) ooa. fi may be placed under the form b cot ^ sin.
fl-l-m COS. I3=c sin. 0-i-m cos. 3, where c representa the height CE of the point
agninsl which the prop rests.

, Google

> BY SHORES.

shore not being driven so as to increase the thrust sustained
by^ it beyond that just necessary to support the wall.*
This least thrust is represented by the foiinula

f^_J^\hmi.a.—{'k-k-^)c,06.a\ -— -^fi-a'A

The thrust which must be given to the prop in order that
there may bo given to the wall any required stability, deter-
mined by the arbitrary constant m, is determlued by equa-
tion (381). The stability will diminish as the value of m is
increased beyond ^», aad the wall will be overthrown
inwards when it exceeds a.

i96. The stability of a wall 6

shore m the sameplaiie.

Let EF, ef be sliores in the same plane, sustaining the
wall ABOD, and both necessary to
its stability; so that if EF were re-
moved, the wall would turn over upon
f, and if ef were removed, upon some
point between F and 0.

If the thrust of the shore EF be
only that just necessaiy to sustain
the tendency of the wall to overturn
upon f it is evident that the hne of
resistance must pass through that
point ; but if the thrast exceed that
just necessary to the equilibrium, or
if the shore be driven then the line
of resistance will intersect f^ in some
point OS. Let/iC=TO ; then represent-
ing the thrust upon EF by Q, the dis-
tances /D and fi by h and i, and tlie angle EFC by ^, the
value of Q is evidently determined by equation (381).

If s be taken in like manner to repr^ent the point where
the line of resistance intersects the base of the wall, and
Cs=m,, OE=ft.; 0=J„ Cfe=/3„ CD=A„ the thrust upon
the prop e^by Q, and its weight by 2w^; then the sum of
tlie moments about the point s of Q and Q„ and the -v

ion of the principle of le

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390 THE STABILITY Of A GOTHIC STKUCTIJEE.

li.ah, of the wall, ecLuals the sum of the moments of P, w^
and 10, ; or

Q,{h+m,) COS. /3, + Q {b,+m,) cos. ^+i^ah, {ia—m,)=
PSA, sin. a—{h+^a—m,) cos. a] + {w+w,)m, (382,)

Substituting the value of Q in this equation, from equation
(381), and solving in respect to Q„ the thrust upon the prop
ef will be determined, so that the stability of the wall, upon
Its section fg and upon its base OB, may be m and m,
respectively.

If »?ij=m, the portions of the wall above and below /gr
are equally stable.

If in,=m=0, the thi-mt upon each shore is only that
which is just necessary to support the wall, or which is pro-
duced by its actual tendency to overtui'n. In this case we
have

im') {h,h-hh,)+T {\-b) {&+^) cos, i
U, COS. /3,

the value of h being detennined by equation

29T. The stahiUl/y of a atructwe havmg parcJM waUs, one
of which is stjypoTt&d iy means of struts resting on tha
summit of the other.

Lot AB and CD be taken to represent the walls, and EP
one of the struts ; the thrust Q upon the
stmt may be determined precisely as in
Art 295. So that the line of resistance
may intersect the base of the wall AB at
a given distance m from the extradoa
(see note, p. 388,)

Let m, represent the distance Die from
the extrados at which the line of resist-
ance intersects the base of the wall CD;
then taking the moments of the prepares
applied to the wall CD about the point
X, as in Art. 295, and observing that
besides the pressure Q the weight w of
one half the stmt is applied at E, we
have
Q|A, sin. ^ + (J,-K— «K) cos. ;3j =f*,c,A (ia,— niO +

,y Google

THE STABILITY OF A GOTHIC STBUCTtlEE. 3111

in whicli equation h, and a, are taken to represent tho
height and tnicknesa of the wall CD, h, tiie distance of the
point E on which the strut reste from the axis of the wall, i3
tlie inclination of tlie strut to the vertical, and (j., the weight
of a cubic foot of the material of the wall.

Substituting for Q its value from equation (381), and
reducing,

Fj/tsin.a— (^+^a) cos, aj — ■^ng%+m(P cos. a + v-ak+w)_
0 sin. /3 +??i cos. /3 ~

A, sin. ,y — (^,+-Ja,— OT,) cos. /3 ^ ''

By this equation is determined that relation between tho
dimensions of the two walls and the amount of the insistent

pressure P, by which any required stability may be a „

to each wall of the structure. If m=0, the pressure upon
the strut will be that only which is produced by the ten-
dency of AB to oveitarn ; and the value of m, determined
from the above equation will give the etabihty of the exter-
nal wall on this supposition.

If m.=0 and «ii=0, both walls will be upon the point of
overturning, and the above equation will express that rela-
tion between the dimensions of the wall and the amount of
the insistent pressure, which corresponds to the state of the
inetabihty of the structure.

The conditions of tho stability, when the wall AB is sup-
ported by two stmts resting upon the summit of the wall
CD, mav be determined by a method similar to the above
(see Art. 296).

The general conditions of the stability of the structure
discussed in this article evidently include those of a Gothic
Btjildiko having a central nave, whose walls are sapported,
under the thi'ust of its roof, by the rafters of the roof of its
side aisles. By a reference to the principles of the preceding
article, the discussion may readily be made to include the
case in which a further support is given to the walls of tlio
nave hyfiymg huUresses, which spring from the summits of
the walls of the aisles. The mfluence of the buttresses
which support the walls of tho aisles upon the conditions of
the stability of the structure forms the subject of a subse-
quent article.

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THE WALL OF A DWELLITTfl.

i8. The stability of a wall sustcdnvng th&jkiors of a

The joists of the floors of a dweUing-lioiise rest at their
extremities tipon, and are sometimes
notched into, pieces of timber called
wall-platee, -which are imbedded in
the masonry of the wall. They
sei-ve thus to bind the opposite sides
of the house together ; and it is upon
the support which the thin walls of
modem houses receive from these
joists, that tiieir stability is some-
thnes made to depend.*
^^ Kepresentiuff by 10 tlie weight of
that portion <rf the flooring which
resta upon the portion ABCD of tlie

/ gH wall, and the distance BE by 0,

/ K taking ic, as before, to represent the

— 1^ point where the line of resistance

intersects the base of the wall, and
measuring the moments from this
point, we have

zE .Cl + xK..iiah + x5.w=xK.'P;

whence, taking the same notation as in the preceding arti-
cles, and substituting,

cQ + (^a— my.fflA+(a— m.)w={A8in.a— (J+^ffl— m)cos.a}P;

:,(^G=\h sin, a~{k+^a) cos. aj P— J^ts'A— wa+

m(Peos. a+^aA+w) (384);

from which expression it appears that Q is less as m- is less.
When, therefore, the strain upon the joints is that only
which is just nec^sary to preserve the stability of the wall,
or which it produces by its tendency to overturn, then
m~0. In th]s case, therefore.

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A WALL SDPPOBl'iDD BY ]

jj^shi. a—(h+ia) COS. al'P^^im'h—wa

. (385).

If /3 be assumed a right angle, and if {a—niyw be substi-
£. tuted for -m/w, tlie case discussed in Art. ^95. will
,/ *' evidently pass into that which is the subject of
the present article, and the preceding equation
may thus be deduced from equation (381) (see
lote, p. 388.).
In like manner, if the wall sustain the pres-
- sure of two floors, and h be taken to represent
the distance from its summit to the lower floor,
and A, its whole height ; then, representing by m
and »ii the distances from the exti'ados at which
the line of r^istance intersects tlie sections EG-
J and eg, and subsdtutina (w + w,) {a — m^ for
. {lo+Wyjm,,, the value of the strain Q on the
joists of the lower floor may be determined by
equation (382), it being obsei-ved that for the
coefficient of Q, in that equation must be substi-
tuted (as was ^own above) the height (A,— A) of
the lower floor from the bottom of the wall. If the strain
be only that produced by the tendency of the wall to over-
turn at a and C, then

Q^e=(A— c) (Jn«'— P sin. a)H-
T{h+ia)cos.a+wa—^^^. . . .

The value of Q is determined by equation (385), o being
taken to represent the distance Ee between the floors, n
the joists be not notched into tlie wall-plates, the friction of
their extremities upon them, produced per foot of the length
by the weight which they support, must at least equal Q and
Qt respectively.

199. The staMUty of a wall sw^orted hy piera or
of uniform tMcMess.

Let tlie piers be imagined to extend along the whole

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WALL SUFFORTEIt BT BUTTBEseES.

length of the wall, as explained in Art. 288. :
and let ABCD represent a section of the con:.-
pound wall thus produced. Let the weight of
each cubic foot of the material of tlie portion
ABFE be represented by f^^, and that of each
cubic foot of GFCD by f,, EA=a„ QD = a„
BG=a, AB = A„ CD = A„ distance from CD,
produced, of the point where P intersects
AE=?, as the intersection of the line of resist-
ance wilh CE, OiB^m. By the principle of the
equality of moments, the moment of P aboiit
the point x is equal to the sum of the momenta
of the weights of GC and AF about that point.
But (Art. 295.) moment of P=P \h, sin. a—
il—m) cos, a} ; also moment of weight of AF=
(a^—m + Ja,)Ajffl,fi, ; moment of weight of GC=

:. P {h, sin. a.—{l~'in) cos. a] =((^5— m+|ff J A,a,fi,+

(:^a,-m)A,a,i>., (387).

K the material of the pier be the same with that of the
wall ; then, talking 5 to represent the breadth of each pier,
and c the common distance of the jiiers from centre to
centre (Art. 288.), cci,(J',=5«^„ therefore o(i.,=ifj-,. Eepre-

senting j- by n, eliminating the value of 1^^ between this

equation and equation (387), writing 1^ for (j-,, and reducing,

P(/ti sin. a— Z cos, '»)=iM' ia'h, + 2a,a,h^ + -a^h, \ —

m]Pcos.a.+)A(«.A+-»A) [• • • .(388);

bv which equation a relation is determined between the
dimensions of a wall supported by piers, having a given
stability m, and its insistent pressure P. Solving it in
respect to a^, the thickness of the pier necessary to give any
required stability to the wall will be determined. (See
Appbhbix.)

If a, be a^umed to represent- that width of the pier by
which the wall would just be made to sustain the given
preaBure P without being overthrown; then taking m=0,
and solvuig in respect to o„

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WALL SUPPOBTED BY EU'

y — =— (A, sm.K— ici

/

300. The stabiliiy of a pier or hittress sur-
mounteahy a pinnacle.

LctW represent the weight of the pionacle,
and 6 the distance of a vertical through its cen-
tre of gravity from the edge C of the pier : then
assuming x to be the point where the line of
resistance intersects the base of the pier, and tak-
ing the same notation as before, equation (387)
will evidently become

Pj^i sin, a—{l—m) COS. a\ = [a^-~-m+^a,\h^a^^-^
\^a,—m] h,a4\ + {e~7n)W.

Substituting for ,u-, its value-W'^or-!-, writing n. for \)-,, and

° en °

reducing,

P(/^sin. «-^cos. a)=if>.U:h, + 2afiA+hi,'h} +

W"e-m|pco8. »+"W"+(*(a,A,-|-^iA) j- ■ ■ (390).

If «, represent the thickness of that pier by which the wall
will just be sustained under the pressure, tatiug m^^O, and

solving in respect to «„ a,— —na,Y-+
l/5^iP(A.sin.»-.c„..-)-W4+»^-(^--l)..- . . im.

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:. eUPFORTED BY GOTHIC BTJITKEeSKg.

The Gothic Butteess.

301. In Gotliic buildings the thicltness of

f a buttress is not unft-equently made to vary

at two or thi-ee different heights above its

baee. Such buttress is represented in the

accompanying iigure.

The conditions by which any rec[uirecl sta-
bihty may be assigned to that poi'tion of it
whose base is 6e may evidently be determined
by equation (390). To deteimine the condi-
tions of the stability of tlie whole buttress
upon CD, let the heights of the points Q, a,
and h above CD be represented by h„ A, and
A,; let DE=:a„ DF=:ffi„ 'FQ=a„ Gx=m^;
then adopting, in other respects, the same
notations as in Arts. 2&9 and 300. Since the
distances from x of tlie verticals through the
centres of gravity of those portions of tlie
buttress whose bases are DE, DF, and FO
respectively, are (aj+Oi+^a, — »i'i), (a,-V\a,
— j«,) and (ia,— wtj) we have, by the equality
or moments,

P|/t, sin, a— (^—mj) COS. aj ^(a^-i^a^^^a^—m'^h^aj^i--^

This equation estabhshee a relation between the dimen-
eions of the buttress and its stability, by which any one of
those dimensions which enter into it may be so detenniued
as to give to m, any required value, and to the structure any
required degree of stability. (See Appendix.)

It is evident that, with a view to the greatest economy of
the material consistent with the given stability of the but-
tre^ the stability of the portion which rests upon the base
he dionld equal that of the whole buttress upon CE ; the
value of TO, in the preceding equation should therefore equal
that of m in equation (S90) If m be eliminated between
these two equations, it being observed that A, and A, in equa
tion (390) are represented by A,— Aj and \—h, in equation
(392), a relation will be established between a„ a^, a, h„ A„
A„ which relation is necessary to the greatest economy of

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f WALLS BUBTAIHINa 1

material ; and therefore to the greatest stability of the strac-
ture with a given quantity of material.

1

The Stabilitt of Waj

802. Thrust -wpon, the feet of the rafters of a roof, the tie-
beam not bainff suspended from the ridge.

If H-, he taken to represent the weight of each sqnare foot
of the rooiing, 2L tlie span, t the
inclination BAG of the rafters to
the horizon, q tlie distance hetween
each two principal raftera, and a,
P" the inclination to the vertical of
' the resultant pressure P on the
foot of each rafter ; then will L see. t represent the length of
each rafter, and f*,!^ sec. t the weight of roofing borne by
each rafter. Let the weights thus home by each of the
rafters AB and BO be imagined to be collected in two equal
weights at its extremities ; the conditions of the equiUbnuni
will remain unchanged, and there will be collected at B the
weight supported by one rafter and represented hj \>-^q
sec. (, and at A and 0 weights, each of ■which is represented
by ^J^q see. t. Now, if Q be taken to represent tlie thmst
produced in the direction of the length of either of the
rafters AB and BO, tlien (Art. 13.) 9'J^ see. i = 2Q cos.
JABC: but ABO = * — 2i; therefore cos. JAJ^O — sin. i;
therefore 2Q sin, i^ti-J^q sec, t ;

_ sect _ fj-.Lg _ i*iLg
■ ' -^2 ain. ( "" 2 sin. ( cos. t sin. 2i'

The pressures applied to the foot A of the rafter are the
thrust Q and the weight ^i^J^q sec. t ; and the required pres-
sure P is the I'esultant of these two pressures. Kesolving Q
vertically and horizontally, we obtain Q sin, i and Q cos. (,
or ^V'J-iq sec. I and ^,Lj coaec. t. The whole pressure applied
vertically at A k tlierefore repi'esented by fi.Lg sec. (, and
the whole horizontal pressnre by ^fi-.Ly cosec, i ; whence it
follows (Art. 11.) that

P = +'fJvf7^°lecr°r+ifi7I?2°^oseer^=
^:^,I2 sec. ( Vl+icot.'t (393).

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KAFTEE3 OF i

tan.a=--''/ ■" ' =^cot.f (394').

(A^Lg sec. t " ^ '

If the incHnation ( of the roof be made to vary, the span
remaining the same, P will attain a minimum value when

tan. t =;— -, or when

4/3

(=35" 16' (395).

It is therefore at this inclination of the roof of a given
span, whose ti-ussee ai-e of the simple form shown in the
iignre, that the least pressm-e will be produced upon the feet
of the raftei-s. If? represent the limiting angle of resistance
between the feet of the rafters and the surface of the tie, tlie
feet of the rafters would not slip even if there were no mor-
tice or notch, provided tliat a were not greater than <f (Art.
141.), or ^ cot. ( not greater than tan. ?, or

( not lees than cot.-^(2 tan. p)* (896).

303. Tfi,e thrust upon the feet of the mft&rs of a roof m
which ihe tie-iea/m is suspended from the ridge iy a

It will be shown in a subsequent portion of this worit
(see equation 658) that, in this case,
the strain upon tne Idng-post BD is
equal to ftSs of the weight of the

p tie-beam with its load. Kepresent-

■i^ ■ ■ Y 'p ing, therefore, the weight of each

j I I foot in the length of the tie-beam
by t^j, and proceeding exactly as in
the last article, we shall obtain for tlie pressure P upon the
feet of the I'afters, and its inclination to the vertical, the
expressions
P=^Li(3i^,2sec.t-l-^^)° + (^28ee.(-|-'(f:,)'cot.''i}* (397).

tan.a=:cot. tU-i^ -M- (398).

* If the surfaces of contact be oak, und thiD alips of oak plank be fixed
under the feet of the rafters, so that the surfaces of contact may present par-
allel fibres of the wood to one anotliec (by which arrangement the Mellon will
be greatly increased), tan. «='48 (see p. 188,); whence it follows that tho
rfiftera ivill net slip, prorided that theii inclination exceed cot."' 'Se, or

, Google

WALL SUSTAnmfG THE THBU3T OF A 3

904. The stahiUty of a -waU sustaining the thnistqfaroof
having no tie-lea m.

Let it be observed, that in the equation tu tlie line of
resistance of a wdll (^ei^iiation
STT), the tenng P sin. a and P
COS. a. represent the horizontal
and vertical pressures on each
foot of the length of the aununit
of the wall ; ami that the fonner
of these pressinea is represented
in the case of a roof (Art. 303.)
by ^n,L cosec. t, and the latter
by (*,L sec, ( ; whence, suhstitu-
ting these values in equation
(377), we obtain for the equation
to the line of resistance in a wall
sustaining the pressure of a roof,
without a tie-beam

y=L

^x cot. (

1—1 «a;cos.(+L

in which expression a rejjresents the thichne^ of the wall,
% tlie distankie of the feet of the rafters from the centre of
the Bummit of the wall, L the span of the roof, (* the weight
of a cubic foot of the wall, and (>-i the weight of each square
foot of tlie roofing. The thickness d of the wall, so that,
being of a given height h, it may snatain the thrust of a
roof of given dimeosions with any given degree of stability,
may be determined precisely, as "in Art. 293, by substituting
A for ai in the above equation, and ^a—m for y, and solving
the resnlting quadratic equation in respect to a.

If, on the other hand, it be required to determine what
must be the inclination i of the rafters of the roof, so that
being of a given span L it may be supported with a given
degree of stability by walls of a given neight A and thick-
ness a ; then the same snbstitutions being made as before,
the resulting equation must be solved in respect to t instead

of ffl.

The value of a admits of a minimum in respect to the
variable *. The valne of (, which determines such a mini-
mum vahie of a, is tliat inclination of the i-afters wliicli is

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BTABILIl'T OF A WALL.

consistent with tlie g:reatest economy in the material of the
wall, ite stability being given.

305. The siahility of a wall supported l>y litttresses, and
eusttmmig the^ressure of a roof without a Ue-hemn.

The conditions of the stability of euch a wall, when sup-
ported by buttresses of -uniform thickness, will evidently be
determined, if in ec[\iation (388) we substitute for P cos. a
and P sin. a, their values f*,L see. t and ipjL consec. i ; we
fihall tlius obtain

FJ.,L (JAi cosec. I— I sec. t)=^\i- {ayi,^+2a,aJi,-\—at,'h^)-
I |J.,L see. i+i^ iajh, + - aji, \ (400).

From which equation the thictne^ «, of the buttresses
necessaiy to give any required stability m to the wall may
be determined.

If the thickness of the buttresses be different at different
heights, and they be surmounted by pinnacles, the con-
ditions of the stability are similarly determined by substi-
tuting for P sin. a. and P cos. a the same values in equations
(390) and (892).

To determine the conditions of the stability of a Gothic
building, whose nave, having a roof without a tie-beam, is
supported by the rafters of its two aisles, or by flying but-
tresses, which rest upon the summits of the walls of its
aisles, a similar substitution must be made in equation (383).

If the walls of the aisles be supported by buttresses,
equation (383) must be replaced oy a similar relation
obtained by the methods laid down in Arts. 299 and 301 ;
the same substitution for P sin. a and P cos. a. must then be
made.

30tl. The conditions cf the staUUiy of a wall supporting a
shed roof.

Let AB represent one of the rafters of sucli a roof, one ex-

,y Google

STABiLrrY r

iOl

trernity A resting against the face of
the wall of a building contiguous to
* ^— -^^iSE, *^^ elied, and the other B upon the
I r ..^^^ ~^ summit of the wall of the shed,
7 „*«^^^ i ^* '^ evident that when the wall

'^'i^^l.. ?EI eh is upon the point of being over-

thrown, the extremity A will be upon
tbe point of slipping on the face of
the wall DC ; so that in this state of
the stability of the wall BH, the direc-
tion of tbe r^aistanee K of the wall
DO on the extremity A of the rafter
wiU bo inclined to the perpendicular AE to ite surface at an
angle equal to the limiting angle of resistance. Moreover,
this direction of the resistance iR which corresponds to tbe
state bordering upon motion is common to every other state ;
for by tbe principle of least resistance {see Theory of the
Aroh) of ^1 tbe pressures which might be supplieQ by tlie
resistance of tbe wall so as to support tbe extremity of the
rafter, its actual resistance is tlie least. Now this least re-
sistance is evidently that whose direction is most nearly ver-
tical ; for the pressure upon the rafter is wholly a vertical
pressure. But the surface of the wall supplies no resistance
whose direction is inclined farther from the horizontal line'
AE than AR ; AR is therefore the direction of the resist-
ance.

Resolving E vertically and horizontally, it becomes R sin,
9 and R COS. <p- Representing the span BF by L, tlie incli-
nation ABF by (, tbe distance of the rafters by j, and the
weight of each square foot of roofing by H-, (Art. 10.), R sin.
ip + P COB. a=f<', Lj sec. I and Rcos. ip~Psiu. a=0; also the
perpendiculars let fall from A on P and upon the vertical
through the centre of AB, are represented by

L COS. (a+i) sec. I and ^L ; therefore (Art. 7).
PL COS. (a+i)sec. (=iL . Lfi, q sec. t, and hence
P COS. (a + ()=^Lfj., y. Eliminating between these equa^
tious, we obtam

=tan. (p + 2 tan. (

(401);

ein.(9 + ()'

P=il>,}-

COS.* (tail. ?4-tan. ()
26

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THE PLATE BAXDE,

If the rafter, instead of resting at A
against the face of the wall, be received
into an aperture, as shown in the figure,
BO tliat the resistance of the wall may be
applied npon its inferior sufaee instead of
at its extremity : then drawing AE per-
pendicular to the surface of the rafter,
the direction AE of the resistance is evi-
dently inchned to that line at the given
limiting angle 9. Its inclination to thehori-

^ *" zon is therefore represented by 5—*+?.

Substituting this angle for <? in eq^uations (401) and (403),
cot.a=cot.(£-'p)+3tan.( (403).

11=

iLfi-,g s

-iLf^iqi^

(i04).

J. ((— ip) + sin. (t— 9)tan. I
Jl + [cot.(t-<p) + 2tan.t]'l*
" t{cot.((— tp) +tan.(^

Sabetitnting in equations (377) and (379) for P sin. a, P cos. a,
their values determined above, all the conditions of the sta-
bility of a wall siipporting such a roof will be determined.

307. The i

il BANDE OK

Let MN represent any joint of
the plate bande AECD, whose

?(jints of support are A and B ;
A the direction of the resistance
at A, WQ a vertical through the
centre of gravity of AMND, TR
the direction of the resultant pres-
sure upon MN ; the directions of
TR, "WQ, and PA intersect, therefore, in the same point O.
Let OAD=a, AM=x, MR=v, Ar)=H, AE=2L, weight
of cubic foot of material of arch=Hi. Draw Rm a pei-pen-
dicular npon PA produced; then by the principle of the
equality of moments,

R^ . P=MQ . (weight of DM).

./Google

THE PLATK EANDE.

403

But Itwi' = (B COS. a, — y sin. k, ]VIQ = ^, weight of DM^
Hiiisr ; also resolving P vertically,

PcoB.oi^LHH'. (405).

Whence we obtain, by substitution in tlie preceding equa-
tion, and reduction,

'L{x—y%w.a)=^^ (406),

which is the equation to the line of resistance, showing it to
be a parabola. If, in this equation, L be substituted for ai,
and the corresponding value of y be represented by T, there
will be obtained the equation T tan. a = ^L, whence it
appears that a is less as x is greater ; but by equation (405),
P is less as a is less. P, therefore, is less as Y is greater ;
but Y can never exceed H, since the line of resistance can-
not intersect the extrados. The least value of P, consistent
■with the stability of the plate bande, is therefore that by
which Y is made equal to H, and the line of resistanee
made to touch the upper surface of the plate bande in F.

Now this least value of P ie, by the principle of l6as6
resistance (see Theory of the Arch), the actual value of the
resistance at A,

.■.tan.a=i2 (*07).

Eliminating a. between equations (405) and (407),

P=LH^j/l+i^, (408).

Multiplying equations (405) and (407) together,

Psin.a=iLV, (409).

Kow P sin. (X represents the horizontal thrust on the point
of support A, From this equation it appears, therefore, that
the horizontal thrust upon the abutments of a straight arch
is wholly independent of the depth K of the arch, and that
it varies as the square of the length L of the arch ; so that
the stability of the abutments of such an arch is not at all
dinainished, but, on the contrary, increased, by increasing
tl^e depth of the arch. Tliis increase of the stability of the
abutment being the necessary result of an increase of the
vertical pressure on the points of support, accompanied by
no increase of the horizontal thrust upon them.

./Google

THE I'Li^TE BANDS.

308. The loaded plate hande.

It 18 evident that the eftert of a loadmg, distributed
uiiitoimly over the extrados of the
jjlate bande, upon its stabiUty, is in
everj rs'^pect the same as would he
piodnoed if the load were removed,
and the weight of the material of
the bande increased so as to leave
the entue weight of the structure
unchanged. Let Hj represent the
M eight of each cubic foot when thus
increased, jj-, the weight of each
cubic foot of the load, and H, the height of the load ; then

. (410).

The conditions of the stability of the loaded plate bande
are determined by the substitution of this value of fi-^ for h-,
in the preceding article.

309. Conditions necessary that the voussoirs of a plate ha/nde
maiy not sUp vpon one another.

It is evident that the inclination of every other resultant
pressure to the perpendicular to the surface of its coiTes-
ponding joint, is less than the inclination of the resultant
pressure or resistance P, to the
: perpendicular to the joint AD.
If, therefore, the inclination he
not greater than this limiting an-
I gle of resistance, then will every
I other coiTesponding inclination
\ he less than it, a

SJ will therefore slip upon the sur-
face of its adjacent voussoir. Now Uie tangent of the incli-
nation P to the perpendicular to AD is represented by cot. a

or by -v.- (equation 407) ; the required condition is therefore

determined by the inequality,

!5<tan.9 (411).

./Google

THE SLOPING BU'lTEESS. 405

It is evident that the liability of the arcH to failure by the
sHpping of its Toussoii-s, is less as its depth is less as com-
pared to its length. In order the more effectually to pro-
tect the arch against it, the voussoirs are Bometimes cut of
the forms shown by the dotted lines in the preceding figure,
their joints converging to a point. The pressures upon the
points A and B are dependent upon the form of that portiou
of the arch which lies between those points, and indepen-
dent of the forms of the voussoirs which compose it ; these
pressures, and the condition of the equilibrium of the piers
which support the ai'cli, remain therefore unchanged by this
change in the forms of the voussoirs,

810. To detennine the conditions of the equiUbiium of
the iipright piers or columns of masonry which form the
abutments of a straight arch, supposing them to be termi-
nated, as shown in the figure, on a different level from the
extrados CD of the arch, let h be taken to represent the
elevation of tlie top of the pier above the point A ; then will

& tan. a, or ^ t? (ecLuation 407), represent the distance AG

(p. 383), or the value of i— -Ja). Substitutingfor k in equa-
tion (377) and also the values of P sin. a, P cos. n, from
Ciquations (409) and (405), we have

X- [h + ^a]

^^^^'^ ^ ^*1^);

which is the equation to the line of resistance of the pier, a
representing its thickness, h the height of its summit above
the springing A of the arch, L tlie length of the ai-ch, n the
weight ot a cubic foot of the material of the arch or ahnt-
ment (supposed the same).

The conditions of the stability may be determined from
this equation as in the preceding articles. If the arch be
uniformly loaded, the value of fi, given by equation (410)
must be substituted for fj-,.

311. The centre of gkavttt of a i

ake incuned at any angle to the veettoal.

let the width AB of the buttress at its summit be repre-

,y Google

THE BLOFHJ& BimBESB.

eented "by a, its width CD at the base by h
its vertical height AF by e, the inclination
of its outer face or extrados BC to the
Tertical by a„ that of its intrados AD by

Let H represent the centre of ^avity of

^ the paralleloOTam ADEB, and K that of

1 the triangle BCE, and G that of the but-

■ trees ; draw HM, GL, KN, perpendiculai-s

upon AF, Then representing GL by >-,

and observing that the area ADEC is represented by ao,

the area EBCfby ^(5-a>f, and the area ADOB by i{a+b)c,

^ .tc.HM+K^-a)cKN_2arHM + (6-fl)KH

Now B.M.='Rh+hM=ia+ic tan. a,=^{a + e tan. «,),
KN=KZ + ;A+AN=|i(5-a)+a+|(J tan. a,=
}{i + 2a+2Gta,n. a.,);
Substituting these values and reducing,

_(a' + tt5 + 5') + (ffl-fM)etan. tt, .

S(a+b) •■■■ ^^^'^''

&=CD=OF-DE=otaii. ct, + »-c tan. a, ; also (»= + «& + &')
=(J— ay+3a5=(r'(tan.a,— tan.a.,)' + 3»o(tan.a,— tan.aj) + 3a ,
(a+35)otan. «,= J2o(tan. a,— tan.a.j)+33} o tan. a^

=2o° (tan. a, — tan. a,) tan. a;+Zae tan, a, ;
.'.{a' +ab+b') + {a+^i) c tan. a,=o' (tan."*,— tan. V,)
+ Sao tan. a^ + 3a''.
, - j^__ jg'ftan. \— tan. '«,)+ ""^ -"-"'

e(taii. a,— tan, Hs) + 2

. (414).

312.

1 Line of

Let LM represent any horizontal section of the buttress,
TK a vertical line through the centre of
gravity of that portion AMLB of the but-
tress which rests upon this section. Pro-
duce LM to meet the vertical AE in V,
and let KV=X and AV=i» ; then is the
value of '>< determined by substituting o)
for c in equation (414)_. Let PO be the
direction m which a single pressure P is
applied to overturn the buttress. Take

* This equation is, of oonrao, to be adapted to the case in wliicli the incUni-
tion of AD is on t!ie other side of tlie yertioal, as shown by tlie dotted line
Ad by making as, and ttierefere tau. nj negative.

, Google

THE SLOPIN& BUTTKESS. 401

OS to represent P in magnitude and direction, and ON tc
represent tlie -weight of tlie portion AMLB of tlie buttress ;
complete the parallelogram SN, and produce its diagonal
OR to Q ; then will OR evidently be the direction ot the
resultant pressure upon AMLB, and Q a point in the line of
resistance.

Let yQ.=f, AQ^k, /GOT=t, fi.=wei_glit of each cubic
foot of material ; and let the same notation be adopted in
other respects as in the last article. By similar triangles,

OK- 01

QK^QV-KV=^-X,

OKt=TK-TO=TK-TG cot. GOT=iB-(x+&) cot. i,

KI=EN sin. ENI^P sin. (,

OI=ON+NI=i(J.AV(AB+LM)+IlN cos. ENI=
-i^ia!|2a+iB(tan. a,— tan. a,)! +Pco3. t;
7— X P sin. (

"fl.~(X + J) cot.

I ^iJ^l^a +x{tan. a^—tm.

s)HPcos..-

Ti'ansposing and

reducing,

i>^{2a+a

1 (tan. tt,— tan. a,)\ +P (a; siu

1. l — 'k COS. I

^^t(cj2ffi+«(tan. a,— tan. a^\ +P cos. i

but substituting x for c in equation (41i), and multiplying
botli sides of that equation by the denominator of the frac-
tion in the second member, and by the factor ^a;, we have

^y^ce{2a+x (tan. a,— tan. a,)\ =ifKc' (tan.v,— tan. '«,) +
ilK^^a tan. a,+i(«M° ; :.'i/=

i«a^(ton.'a,-tiin.'a,)+^'ato.«.+j/m°+2P(j^em.t-i;coa-0 .

which is the equation to the line of resistance in a
If tlie intrados AD he vertical, tan. a, is to be assumed =0,
If AD be inclined on the opposite, side of the vertical to that
shown in the figui'e, tan. a, is to be taken negatively. Th&
line of r^istance being of three dimensions in ». it follows
that, for certain values of y, there are three possible values
of X ; the curve has therefore a point of contrary flexure.
The conditions of the equilibrium of the buttress are deter-

,y Google

408 WALL etrSTAIHINS THE

mined from its line of resistance precisely as those of llie
wall.

Thus the thickness a of the buttress at its summit being
given, and its height c, and it being obeei-ved that the dis-
tance CE is represented by a-i-c tan. a„ the inclination a, of
its exfcradoB to the vertical may be determined, so that its
line of resistance may intersect its foundation at a given dis-
tance m from its extradoa, by solving equation (415) in re-
spect to tan. ct„ having first substituted a for x and a+o tan.
a,— 7/1 for y ; and any other of the elements determining the
conditions of the stability of the buttr^s may in like manner
be determined by solving the equation (the same substitu-
tions being made in it) in respect to that element.

If E be taken to represent the surface of the fluid, IK any
hcction of the wall, and EP two thirds
the depth EK ; then will P be the cen-
tie of pressure* of EK, the tendency
t the fluid to ovei-turn the portion
\XIB of the wall being tlie same as
\ )i!ld be produced by a single pressui'e
tujplied perpendicular to its surface at
P and being equal in amount to the
weight of a mass of water whose base
IS equal to EK, and its height to the
deptn of the centre of gravity of EK, or

tnJ-EK LLtAK=» AE=e, weight of each cuhic.foot of
tl c flmd— (J.

:.V={x-e).i(x-e)i^,=i{a!-ey^

Let the direction of P intersect the axis of the wall in O ;
let it be represented in magnitude by OS ; take ON to
represent the weight of tlie portion AKIB of the wall ; com-
plete the parallelogram SN, and produce its diagonal to
meet IK in Q ; ^en will Q he a pomt in the line of resist-
ance. Let Qi/L=y, AB=«, weight of each cubic foot of

QM E.X
.material of waU=(J'. Ey similar triangles, Trfn=-w7)' ■^*^^'

ind Hyarodynamicf," by the author of this

, Google

PKE6BUEE OF A FLUID. 409

ii^X'^—^}% NO=weiglit of ABIK=(taic ;

Dividing numerator and denominator of this equation by

[J.,, and observing tliat tlie fraction — represents tbe ratio a

of tbe specific gravities of the material of tbe wall and the
fluid, we have

y=S^^ (416);

wbicb is the equation to tbe line of resistance in a wall of
uniform thietness, snstaining the pressure of a fluid.

314. To deUrmi'M the thickness, a, of the wall, so that its
height, h, lemg gi/oen, the line of resistemoe may interseot
its foundation, at a given dista/iwe, m, withvn, the e^Todos.

Substituting, in equation (416), h for a), and %«— m for y,
and solving tbe resulting equation in respect to a, we obtain

. (lit)

Equation (416) may bo put under tbe form y=
-7. — -^ ( 1 — -j ; whence it is apparent that y increases con-
tinually with a; ; so tliat tbe nearest approach is made by
tbe line of resistance, to the extrados of the pier, at its
lowest section, m, therefore represents, in the above expres-
sion, the modulus of stability (Art, 286).

315. The conditions Tiecesswry that the wall should not he
ov&rth-own by the sUpping of the courses of stones on one

The angle SEO represents the inclination of tbe resultant
pressure upon the section IK to the perpendicular ; tbe pro-
posed condition is therefore satisfied, so long as SEO is less
than the limiting angle of resistance ?,

./Google

Now, tan. SEO:

WALL SUaTAININa THE

- OS KM" iff."-')'.

■"SK-ON-

; the proposed con-

, ("— )■ .

difion is therefore satisfied, so long as —n7^ < t^"- f i or,
reducing tliis inequality, so long as

2e cot i|i\* )

a!<6+(T(i! tan. i

1+ i+-

. (418.)

316, The Stability of a wall of variable thickkess

eUSTAraiSG THE PEEaaUKE OF A ELL'ID.

Let US Urst suppose the internal face AB of the wall to be
^ veilical ; let XT be any section of it,

P the centre of pi-essure of EX, and
_^^ SM a vertical thi-ongh the centre of
1=^^ gravity of the portion AXYD of the
.^fa==== -^vall. Produce the horizontal direc-
tion of the pressure P of the fluid,
supposed to be collected in its centre
of jpreesure, to meet MS in S, and let
SK be taken to represent it in mag-
nitude, and ST to r^jresent ihe ■weight
'■ " of tlie portion AaYD of the wall,

and complete the parallelogram STKK ; then wUl its
diagonal SE represent the direction and amount of the
reaStant pressure upon the maes AXYD, and if it be pro-
duced to mtei-sect XY in Q, Q will be a point in the line of

Let AX=ii!, XQ=j/, MX=x, AE=fi, AD=a, inehnation
of DO to vertical^a, fj.— weight of cubical foot of wall,
(j.,=weight of cubical foot of fluid. By aimilar triangles,
QM ET ^^
SM = 8f- ^""^
QM=QX-MX=y-X, SM=PX=iEX*=i(a;-e);

RT=pressure of fluid on EX:^iEX.t'.EX=J(j.,(6-tf)'t
ST=weiglit of mass AY=^^j3a-|-a? tan. a.\^.

" The centre of preasure of a
pressure of a Quid Is situated at t
Uydrostaties, p. 2B.

I The pressure of a huavj fluid o
of a pibm of the fluid whose base i
its height to the depth of the cei
Hydrostatics, Art. 31.

L an; plane suiface is equal to the weight
equal in area to the surfai e pressed and
tre of graTity of the surface pressed.—

, Google

PiiESSUllE OF A FLGID.

Let— =o; tlien, if the fluid be water, a represents the

specific gi'avity of the material of the wall ; and if not, it
represents the ratio of the specific gi-avities of the fluid and
wall.

Now maldng a,=;0 in equation (41i), and substituting «
for a„ and ic for c,

■Jar' tan. 'a + ffic tan. a + «°_-Jar' tan. V + «ic^ tan. a+ffl'iK
~ ictan. a + 2a ~ ^ax+x'ts.n.. a

Adding this equation to the preceding,

^{iK— ■ey+^'tan.'a+otc' tan. o.+a'x

^= 2ffic+ic"tan.te ' ' ' ' ^^^^^ '

which is the equation to the line of resistance to the wall,
the conditions of whose stability may be determined from it
as before (see Ai-ts. 291. 293.).

317. The conditions necessary ihat no course of stones oomr
posinff the wall may sUp upon tJis subjacent cou?'se.

This condition is satisfied when the inclination of SQ to
the perpendicular to the surface of contact at Q is less than
the limiting angle of resistance <? ; that is, when QSM<?,
or when

ortan.9>l-)2^_^^^^^_^

No course of stones will be made by the pressure of the
fluid to slip upon the subjacent course so long aa tliis condi-
tion is satisfied.

It is easily shown that the expression forming the second
member of the above inequality increases continually with

./Google

412 THE HATtTBAL SLOPE OF EARTH.

X, BO that the obliquity of the resultant pressure upon each
coTiree, and the prohability of its heiug made to slip upon
the next subjacent conree, is greater in respect to tixe lower
than the upper coursesj increasing with the depth of each
conrse beneath the surface of the Huid,

Eaeth Wokkb.

318. The nainrcd slope of earih.

It has been explained (Art. 241.) that a mass, placed upon
an inclined plane and acted upon by no other forces than its
weight and the resistance of the plane, will just be supported
when the inclination of the plane to the horizon equals the
limiting angle of resistance between the surface of the plane
and that of the mass which it supports ; so chat the limiting
angle of resistance between the surfaces of the component
parts of any mass of earth might be determined by varying
continually the slope of its surface until a slope or inclination
was attained, at which particular slope small masses of the
same earth would only just be supported on its surface, or
would just be upon the point of slipping down it. Now this
proce^ of expenment is very exactly imitated in the case of
embankmente, cuttings, and other earth-works, by natural
causes. K a slope of earth be artificially constructed at an
inclination greater than tlie particular inclination here
spoken of, although, at firet, the cohesion of the materia!
may so bind its parts together as tJD prevent them from slid-
ing upon one another, and its surface from assuming its
natural slope, yet by the operation of moisture, penetrating
its mass and afterwards diying, or under the influence ot
frost, congealing, and in the act of congelation expanding
itself, this cohesion of the particles of the mass is continually
in the process of being desti'oyed ; and thus the particles, so
long as the slope exceeds the limiting angle of resistance,
are continually m the act of sliding down, until, when that
angle is at length reached, this descent ceases (except in so
far as the particles continue to be washed down by the rain),
and the surface retains pennanently its natural slope.

The limiting angle of resistance f is thus determined by
observing what is the natural slope of each description of
earth.

./Google

THE PKESBUKE C

The following table contains the results of some sticb
otservatioiis*: —

P DiFFEREMT KlNI)S 0

Nstute of Barlh,

Naiuial Slope.

Aui,,...,.

Fine dry Band (a single esperiineat) -

Ditto

Ditto

Common eapili pnlverised and dry -
Common earth aliglitl)- damp -
Earth the most dense and compact -
Loose shingle perfectly dry

21°

34° 29'
S9°
48° 50'

65°
89°

Gadvoy.

Hondelet.

Barlow.

Rondelet.

Eondelet.

Barlow.

Fasley.

SfKCIFIC GKiyiTIES o

Nature of Earth.

Epeoiflc GriTily.

Vegetable earth .......

Sandy earth .

1-4

I'9
1-7
1-t to 2-3
2-3
2-5

Earthy sand

Bubble maeoTiry of granite

Bubble masonry of basaltic stones ....

319. The prebsuke of eaeth.

Let BD represent tlie surface of a wall sustaining the
pre^ure of a mass of earth -whose surface AE is horizontal.

Let F represent the resultant of the pressures sustained
by any portion AX of the wall ; and let the cohesion of the
pai'ticles of the earth to one another he neglected, as also
their friction on the surface of the wall. It is evident that

./Google

414 THE PEESSUKK OP EAETH.

any reeulte deduced in respect to
the dimensions of tlie wall, these
,^yK elements of the calculation being
/; neglected, will be in eiecess, ana
|« eiT on the safe side,
j," How the masa of earth which
i§ presses upon AX may yield in the
'* ^™ction of any oblique section
U^ XT, made from X to the surface
^ AE of the mass. Suppose YX to
^ he tlieparticulai'direetioninwhich
it actually tends to yield ; so that
^ if AX were removed, rupture
'''■'^>^^^^- would first take place along tliia
section, and AXY he the poi-tion of the mass wbich would
iirst fall. Then is the weight of the mass AYX supported
by tlie resistances of the different elements of the surface AX
of the wall, whose resultant is P, and by the reBiBtance of
the surface XY on which it tends to slide. Suppose, now,
that the mass is upon the poiut of eliding down theplane
XT, the pressure I* being that only which is just sumcient
to support it ; the resultant SR of the resistances of the
different points of XY is dierefore inclined (Art. 241.) to the
normal ST, at an angle EST equal to the limiting angle of
resistance f between any two contiguous smfaces of the
earth.

Now the proBSure P, the weight W of the mass AXY, and
the resistance K, being pressures in equilibrium, any two of
them are to one another invereely as the sines of their incli-
nations to the third (Art. 14,),

. P_ sin, WSR T>_w sin. WSR
•■W^8in.PS"E ' "^-^ sin.PSR'

But "WSE=WST-RST=AYX-KST=^-i-9,

if AXY=£; PSR=PST+EST=AXY+EST=(+fl).

.•.P="W"cot.(i+<p) .... (421).

Also W=-Jfi,AX . AY=i(t,iK'' tan. ( ; if (J.,=weight of each
cubic foot of eai-th, and AX=a; ;

.■.P^Jj^.k" tan. ( cot. ((+9) .... (422).
Now it is evident tliat this expression, which represents

./Google

EAKTH. in

the resistance of the wall necessary to sustain the pi-essnre of
the wedge-shaped mass of eai'th AXY, being dependent fot
its amount upon the value of ( (so that different sections,
Buch as XT, oeing taken, each different mass cut off by such
section will require a different resistance of tlie wall to sup-
port it), may admit of a maximum value in respect to that
variable.* And if the wall be made sti'ong enough to supply
a resistance sufficient to support that wedge-shaped mass of
earth whose inclination i corresponds to the maximum value
of P, and which thus requires the greatest resistance to sup-
port it ; then will the earth evidently be prevented by it from
slipping at any inclination whatever, for it will evidently not
slip at thai angle, the resistance necessary to support it at
that angle being supplied ; and it will not slip at any other
angle, because more than the resistance necessary to prevent
it Sipping at any other angle is supplied.

If, tlien, the wall supplies a resistance equal to the maxi-
mum value of P in respect to the variable (, it will not be
overthrown by the pressure of the earth on AX. Moreovei-,
if it supply any less resistance, it toiU be overthrown ; there"
not being a sufficient resistance supplied by it to prevent the
earth from sHpping at that inclination t which coiTesponda
to the maximum vSue of P.

To determine the actual pre^ure of the earth on AX, we
have then only to determine the maximum value of P in re-

t?P ^ A^^^^
-^=0, and-.f-5-<0.
at <u

But differentiating equation (422) in respect to i, we obtain
by reduction

g=i.x"°-^'+'>T;'°-?' ■ ■ ■ ■ ("23)+

ai cos. ( sm. ((+?)

Let the numerator and denominator of the fraction in the

e of this maximum will aubsoquentiy be shown: it is, how-
eyer, sufficiently evident, that, aa the angle i Is greater, the wedge-shaped miSS
to be supported isbeavieT; for which cause, if it operated alone, P would be-
come greater ss i inoceased. But as i increases, the plane XT becomes lees
inollned; for whioh cause, if it operated alone, P would become less as i in
creased. These two causes thus operating to counteract one another, deter
mine a certain inclination in respect to which their neutralising influence is the
least, and P tlierefore the greatest.
] Church's Diff. ami Int. Cal,, Art. 41.

./Google

416 llEVETEMENTB,

second memLer of this equation be represented respectively
by -p and j ; therefore -^=\v-^^ . -, 1^? ~;fp) ' ^^^ '^^^^
-^=0,^7=0; inthiecase,therefore,^=ii^,ie'i^. Wbence

it follows, by substitution, that for every value of i by which
the first condition of a raaxinium is eatislied, the second dif-
ferential co-efficient becomes

ai COS. tfiin. (t-i-9) ^ '

Kow it is evident from ectuation (423) that the condition
-=-=0 is satisfied by that value of ( which makes 2((-|-(p)=
ir— 2(, 01"

'=1-1 (^^^)-

And if this value be eubstitnted for t in equation (424), it
nes

=(J.,ic'-

ViSr \4^2)

which expression is essentially negative, so that the second
condition is also satisfied by this value of i. It is that, there-
fore, which coiTesponds to the maximum value of P ; and
substituting in equation (422), and reducing, we obtain for
this maximum value of P the expression

P=i^«=tan.'(^--|j....(427);

which expression represents the actual presRuro of the earth
on a surface AX of the wall, whose width is one foot and its
depth X.

EEVKrEMEST "Walls,
If, instead of a revetement wall sustaining the pres-

./Google

417

sure of a tnass of eartli, the weight

n of each cubic foot of which is re-

'^■B presented by [>-„ it had sustained

I the pressure of a Jkdd^ the weight

of each cubic foot of which was re-

^7 $ presented by f*, tan. ' (7— |) t Aen

■essure of that fliii
?face AX have be€

by j^/i,^tan.= g-|

would the pressure of that fluid
upon the surface AX have been

_j that the pressure of a mass of
earth upon a revetement wall (equation 427), when its sur-
face is horizontal (and when its horizontal surface extends,
as shown in the figurt>, to the verj surface of the wall), is
identical with that of an imaginary fluid whose specific gra-
vity is such as to cause each cubic foot of it to have aweight
i in pounds hy the formula

|.,=M.n.-g-|) ....(428);

Substituting this value for ;>■, in equations (416) and (419),
we determme therefore, at once, the lines of resistance in
revetement walls of uniform and variable thickness, under
the conditions supposed, to he respectively

fc"iZ

■ (429) ;

^^g ^^ ^' (430) ;

■where c represents the ratio of the specific gravity of the
material of the waU to that of the earth, Tlie conditions of
the equilibrium of the revetement wall may he determined
from tlie equation to its line of resistance, as explained, in
the case of the ordinaiy wall.

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321. The conditions necessary that a revetement wall may
not he overthrown hy the slicing of the stones of any
cowrse wpon those of the siihjacewt cov/rse.

These are evidently determined from the inequality (420)
uy substituting f^, (equation 438) for f*, in that inequality ;
we thus obtain, representing the limiting angle of resistance
of the stones composing the wall by ip, to distinguish it from
that <9 of the earth,

tan. 9,>- tan. ' 7— H H— ^^ — rr (^31);

. ' o \i 21 2ax^x tan. a •• /'

where o represents the ratio of the specific gi-avity of the
material of the wall to that of the eai'th.

As before, it may be shown from tliis expression that the
tendency of the courses to slip upon one another is gi'eatcr
in the lower coui-ses than the higher.

822. Ths pressure (f earth whose s-
horizon.

■rface is inclined to tJie

Let AB represent the surface of such a mass of earth, YX
the plane along which the
rupture of the mass iu
contact witJi the surface
AX of a revetement wall
tends to take place, AX=
3!, AXY=(, XAB=/3.
Tlien if W he taken to
represent the weight of
the mass AXT, it may be
shown, as in Art. 319,
equation (421), that P=
W cot. (( + ?)).

X sin. (
AY= .- ■■r-.^X there-

Now "W=ii*,AX.AY.sin.
a? sin. ( sin, ^ ,

sin. ((+/3):

fore W=:i«:,-

sin. (i+iif)

:.v^^^y.

cot, (t+tp)

. (432).

' cot, i + COt. /

Now the value of * in this function is that which renders
it a maximum (Art. 319). Expanding cot. (t+v), and dif-

,y Google

EEVETF.MENTe.

fereiitiating in respect to tan. i, tliie value of t it
determined to be tliat which satisfies the equation

cot. (=tan. (p + sec. <p t^l + cot. /3 cot. ip . . . . (433).
Snbetituting in equation (432), and reducing,

T~ii^,x

(1 + ein. 9 Vl + cot. <i> cot. /3 )

. (434).

From which equation it is apparent, that the pressure of the
earth is, in this case, identical with that of a fluid, of such a
density that the weight /j-,, of each cubic foot of it, is repre-
sented by the formula

J. -^ i "^^-^ 1° ..... (435).

The conditions of the equilibrium of a revetement wall
fiuataining the pressui'e of such a mass of earth are therefore
determined by the same conditions as those of the river wall
(Arts. 313 and 316).

323. The Hesistance of Eaeth.

Let the wall BDEF bo supported by the resistanee of a
mass of earth upon ite sur-
face AD, a pi-eesure P, ap-
plied to its opposite fece,
tending to overthrow it. Let
I the sniface AH of the earth
horizontal; and let Q
fi represent the pressure which,
;| being applied to AX, would
■h just be suifioicnt to cause the
d mass of earth in contact
I with that portion of the wall
ffiwwr.v'^ to yield; the prism AXT
" ■" slipping backwards upon tie
surface XY. Adopting the tame notation as in Art. 319,
and prociiediiig in the same manner, but observing that E8
is to oe measured here on the opposite side of TS (Art. 241),
since the mass of earth is supposed to be upon the point of
elippiug upwai-ds instead of downwards, we shall obtain
Q=-Jfi,a!' tan. i cot. (i~^) (436),

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420

i BACKED BY ]

Now it is evident that SY is that plaiie along which rup-
ture may be made to take place by tJie least value of Q ; t
in the above expre^ion is therefore tliat angle which givea
to that expression its minimum value. Hence, observing
that eqnation (436) differs from equation (422) only in the
Bign of flj, and that the second difierential (eqnation 42C) is
rendered essentially pc«itive by changing the sign of <f, it is
apparent (equation 42T) that the value of Q necessary to
overcome the pressure of tlie earth upon AX is represented

Q=frXt.n.')j+^)

. (43t).

324. It is evident that a fluid would oppose the same
resistance to the overthrow of the wall as the resistance of
the earth does, provided that the weight ^^^ of each cubic
foot of the fluid were such that

[^,=t^,tan.'(^+g) ....(438);

BO that the point in AX at which the pressure Q may be
conceived to be applied, is situated at |ds the distance AZ.

325. Th.6 stability of a wall of imiform thichness which a
given pressure P tends to overthrow, and which is sus-
toMied hy the resistance of earth.

Let y be the point in which any section XZ of the wall
would be intersected by the
resultant of the pressures
upon the wall above tliat sec-
ik,^^.„.ai._,^^„ tioi) if the whole resistance
i,-,:.r„..-.v.>-.^ ^^ Q^ which the earth in con-
3 tact with AX is capable of
i supplying, were called into
}\ action. Lot BX:=a!, X.i/=:y,
I BA = e, BE=a, 'Bp = %,
;i weight of cubic feet of ma-
^ terial of wall=(*, inclination
t of P to verrical=5. Taking
'"' the moments about the point
y of the pTessurcs applied t« BXZE, we have, by the prin-
ciple of the equality of moments, observing that XQ— ^

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(ic—^), and that the pGi-pendicular from y, upon P ia repre-
sented by X ein, 6—{k^y) cos. 6,

Pjte sin. i~{h—y) cos. 6\ =^{x~e)Q + {ia—yyax;
or aubstitating for Q its value (equation 437), and solving in
respect to y,

_^f*X^— 6)'+^!J-o'iK— P(a! sin. 6—'k cos. 6)

^~ P COS. S +f*«iB ^ ''

Now it is evident that the wall will not be overthrown
upon any sectioa XZ, so long as the greatest resistance Q,
■wrhich the enperincumbent earth is capable of supplying, ia
sufficient to caoee the resultant pressure upon EX to inter-
eect that section, or bo long as y in_t±ie above equation has
a positive value ; moreover, thai the stahiUiyy of the wall is
determined Tyy the minimum value of y in reject to x in
that equalion, a/yid the greatest height to which the wall can
le huUt, so as to stand, hy that value ofsswhiohtnakea y=0.

526, The stabiUty of a wall which a given pressv/r-e tends to
overthrow, wnd which is supported hj a mass of earth
whose surface is not horiBontat.

Let ^ represent tlie inclination of the surface AB of earth
to the horizon. By reasoning
similar to that of Art. 323., it is
appaj'ent that the resistance Q
of the earth in contact ■with any
given portion AX of the wall to
displacement, is determined by
assigning to p a negative value
in equation (434). Whence it
follows, that this resistance is
equivalent to that which would
be produced by the pressure of
,.^, a fluid upon the wall, the weight
^&^8f-tfa&H»tfg.^!i\5 fi^ of each cubic foot of which
was represented by the formula

1. ip Vl— cot. ip cot. /3)

. (440).

The conditions of the stability of an upright wall sub-
jected to any given pressure P tending to overthrow it, and

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siistained by the pressure of such a mass of earth, are there-
lore precisely the same as those discussed in the last article ;
the symbol i*, (equation 439) being replaced by fi, (eciuation
UO).

327. The stahiKty of a reKet&ment waU whose interior face
is incUned to the vertical at any angle ; taking into account
the friction of the earth v^pon the face of the -waU.

Let a, represent the inclination of the face BD of such a
waU to the vertical, ip, the limiting angle of resistance
between the mass of earth and the surface of the wall ; and
let the same notation be adopted as in the last article in
i-espeet to the other elements of the
question, and the same construction
' made. DrawPQpei-pendiculai-toBD;
then is the direction PS of the resist-
ance of the wall upon the mass of earth,
evidently inclined to QP at an angle
QPS equal to the limiting angle of
resistance 9,, in the state bordering
upon motion by the overthrow of the
wall* (Art. 241.).

Draw Pn horizontally and Xa verti-
cally, produce T8 and BS to meet it in
m and n, and let ((XY=i,

P sm. WSK sin. ("WST-TSR)
■*• W- sin.-PSP~8in.(,fiTOP + SP«i}.

But "W"ST=AYX=^-«.XY=|-£,TSE=?,

RmP=TwP+mSn==oXY+EST=(-|-?,
SPTO = SPQ + QPn=<p,+a,:

^ p_ sin ^^-^-yj ^ cos.(t+a)

"■^"smT^r+^+vW sin.(i-l-a,+'p+'P,)'

Also W^^^i.,'^ . AT=4iJ.,ic' (tan. i+tan. «,) ; if a'K.=x,

* It is not only in the state of the wall bordering upon motion that this
direction of the re^stance obtains, but in ETerj state in whicli the stability of
the wall is maintained. (See the PnnicipU of Least EeaUtance.)

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EEVETEMENT8. 428

(441).

,coB.(t + 9)(taii.f+taii.ii,)
sin.{i + a, + v+f,)

A,EBUining it, + f+9,=0, then difl'erentiating in respect to *,

dP
and assuming -^ = 0, we obtain by reduction

— (tan. (+tan. a,) cob.{/3— 9) +

cos.((+'p)Bin.((+^8ec.'t=0; or,

— (tan. I + tan. a,) (1 + tan. ^ tan. f>) +

(1— tan. i tan. ?) (tan. i+tan. /3)=0 ;

.-. tan." t + 2 tan. i tan. ^ — tan. /3 cot. <p +

(cot. p + tan. /3) tan. «, = 0.

Solving this quadratic in respect to tan. i, neglecting the

negative root, since tan, t is essentially positive, and reducing,

tan. (=(tan. ^— tan. «,)Ktan. i^ + cot. 9)*— tan. /3 . . . (442.)

ITow the value of ( determined by this equation, when

substituted in the second differential coefficient of P in

respect to i, gives to that coefficient a negative value ; it

therefore corresponds to a maximum value of P, which

maximum determines (Art. 319.) the thrust of the earth

upon the portion AX of the wall. To obtain this maximum

value of P by substitution in equation (441), let it be

observed that

COS. (f+y)_l— tan, t tan, y /cos, y \

sin. (t+/3)~ (tan. t+tan. ^) Uos. ^V

1— tan. t tan. ip=H-tan. /3 tan. ¥>— tan, v (tan, /3 —

tan. a,)^(tan, /3+cot. 9)+,

=tan. (p (tan. ^ + cot. 9)* {(tan. ^ + cot. p)^— (tan. /3— tan. a,)*f

tan. t+tan. ^=(tan. /3 + cot. ip)Ktan. ^— tan. a,)*;

COS. {i + <?) sin, tp i /tan, g + cot. yU )

"sin. ((+/3)-coB, S 1 I tan. ;8— tan. « j ~ "^ i

Also tan. i-i-tan. K,=(tan. /3-)-cot, (p)'(tan. /3—

tan. a,)^— (tan. /3— tan. a.^

=(tan. ^— tan, aj)S|(tan. /34-cot. 9)*— (tan, fJ— tan. a,)'},

.■.P=^,ai''^^Ktan,/3 + eot.9)t— (tan./3-tan.«,)*i';

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RSVETEMENTS.

■which expression may "be placed under the following form,
better adapted to logarithmic calculation,

'P-il^.x"

sin, y j /COS. {^^•p)\^ /sin. (,3— a,jVi-|
cos, "/S t \ sin. 9 / ~ \ cos. a, I )

or suhstituting for /3 its value a^+ip + 'p,,

^ , (ijaf'sin. (p j /coa. (((, + 92)1^

'^COS-Xtti + ip + ips) I

/sin. (ip+ff,)\* i

sin, (p )
. . (443).

By a comparison of this equation ■with equation (437) it
is apparent, that the pressure of a mass of earth upon a
revetement wall, nnder the supposed conditions, is identical
■with that which it ■would produce if it were perfectly fluid,
provided that the -weight of each cubic foot of that fluid had
a value represented by the coefiicient of ^ in the above
equation ; bo that the conditions of the stability of eueh a
revetement wall are identical (this value being supposed)
with the conditions of the stability of a wall sustaining the
pressure of a fluid, except that the pressure of the earth is
not exerted upon the wall in a direction pei'pendicular to its
surface, as that of a fluid is, but in a direction inclined to
the pei-pendicular at a given angle, namely, the limiting
angle ot resistance.

. The rKESsmtE of eahtii ■which sokmocnts a eevete-.

MENT WAIL AND SLOPES TO ITS SUMMIT.

Hitherto we have supposed the surface of the earth ■whose
pressui'e is sustained by a revete-

/ ^ now suppose its surface to be ele-

/ $ vated above the summit of the ■wall,

/ I and to descend to it by the natural

I slope ; the wall is then said to be

'ii surcharged, or to carry a parapet.

^, Let EF represent the natnral slope

of the eai'th, FY its homontal sur-

BX any portion of the internal

or intrados of the wall, P the

horizontal pressure just necessary

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to support the mass of eai'th HXYF, whose wei^t is W,
upon tiie inclined plane XY. Produce XB and YF to meet
in A, and let AX=ie, AH=:e, AXY=(, F,=weight of each
cnbic foot of the earth, 9 the natural slope of its surface
FE. Now it may be shown, precisely hy the same reason-
ing as before, that the actnal pressure of the earth upon the
portion BX of the wall is represented by that value of P
which is a maximum in respect to the variable t ; moreover,
that the relation of P and i is expressed by the function P
:^W" cot. (i+ip); where "W"=fi.,(,area HX:YF)=^(AXY-
AIIF)=f».,(-i^'tan. t— -J^cot. (p);

.■.P=^.(ic* tan. 1—0' cot. o) cot. ((+?) {44i).

Expanding cot. (i+p),

■p_i (i"' tan, t— e° cot, y) (1— tan, t tan, f)

P— 4(i,— tan.i+tan.9

To facilitate the differentiation of this function, let
tan. ( + tan. ip be represented by s, and let it be obseiTed
that whatever conditions determine the maximum value of P
in respect to s determine also its maximum value in respect
to t.* Then tan. i=s— tan. <? ; therefore 1 — tan. i tan, ip=
1— s tan. <p+tan. \=—s tan. (p + sec. ''<p. Also, as' tan. i—
c" cot, tf=x'3— {s? tan. f+d' cot. f).

Substituting these values in the preceding expression for
P, and redncing,

-„ , ( , (sf tan, a + (f cot, 9) sec. 'o
P=-|(j., I —sx' tan. If— ^^ '- +

»i''(sec, 'a: + tan.V) + <'' [ (i4o),

dV , ! , (si' tan, (p + c* cot. 9) sec. '<? }
:.^ =i^ j -X tan.,+ ^, \ ,

« For -^=-

di~ih di' di' ~ dz^ \di / A tfi' '

foro -T- = -.- sec. 'i; and for all Tables of i less than -, sei
value, so that "t- = 0 when -t-=0.

When, mort^orer, -^-=0, -rr = -i^ T" ) ! so '''"»> "lien
-J— Is also negative.

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4aB EEVETEMENTS.

<fP (at" tan, ip+c* cot, y) sec. V

The first condition of a maximum ia therefore satisfied by
the equation

~-iB''tan.ip+^^ —, ■ =0 .... (446);

or, solving this equation in respect to s, and reducing, it ia
satisfied by tbo equation

Now the second condition of a maximum is evidently
satisfied by any positive value of z, and tlierefore by the
jioeitive root of this equation. Taking, therefore, the posi-
tive sign, substituting ior s its value, and transposing,

tan. t= I sec. V + -j cosec. \ \ —tan. 9 (447) ;

which equation detei-minea the tangent of the inclination
AXY to the vertical, of the base XY of that wedge-like
mass of eai'tb HXTF, whose pressure is borne by the sur-
face BX of the wall. To detenniue the actual pressure
upon the wall, this value of tan. ( must be substituted in the
expression for P (equation 445). Now the two first terms
of the expression within the brackets in the second member
of that equation may be placed under the form

j {x' tan. !f-\-o' cot, ip) sec. \ )

But it appeare by equation (446) that tlie two terms which
compose this expression are equal, so that tlie expression '
equivalent to — Saaj* tau. (p ; or, substituting for the va^

z, to —%a? tan. ip (sec. '9+;^ cosec. '9)^, or to — 2a; se

(as" tan, V 4-e')^, Substituting in equation (445),

P=^H, j— Sksog, ip{a!'tan.V+c')* +(35" tan. >+(;')+«'' sec. V}
.■.P=i^ja!sec. ip-(iB°tan. ''>p+o')*|'' (448);

by which expression is determined the actual pressure upon
a portion of the wall, the distance of whose lowest point
from A is represented by x.

./Google

E15VETKMICNTS.

82(t. TK& oonditions necessary that a revetemeiit wall oarry-
ing a parapet may not ie overthrown, hy the slipping of
any course cf stones on tJie su!yjaeent course.

Lot ipj represent the limiting angle of the reaistanea of the
stones of the ■wall upon one aaoUier ; and let OQ represent
_jr tlie direction of tlie resultant pressure
"/i on the course XZ. The proposed
f I; conditions are then involved {Art.

3 141.) in the ineqnality (p,>QOM, or
'i tan. ip, > tan. QOM, or tan. ip, >

I Di>weighfofEZ»' """"itating
5' for P its value {equation 448), and
'i ^^{Sa^K+iB' tan. a) for the weight of
^ BZ, it appears that the proposed

4 conditions ai'e deteraiined hj the
'.ncquality

-{x'tan.\+cy\'\

.{449}.

330. The line of resistcmce in a revefement waU carrymg a

Let OT be taken to represent tlie pressure P, and OS the
■weight of EZ. Complete the parallelogram ST, and pro-
duce its diagonal OE to Q ; then ■will Q be a point in the
line of resistance. Let AX=x, QX=y, AJi~b, AP=X,

XM^\ ■W = ■weight of BZf. By similar triangles, %u=

ES

os'

but QM=(y->-), OM=iB-X, ES=P, OS=V;

. f-^ 'P

WX + Par-PX

. (450).

Kow the value of X is determined from equation (414), by

* The influence, upon the equilibrium of the ivall, of the small portion of
fftrth BHG is neglected in this and the snbaaquent eomputation.

■f The influGnue of the weight of the small maas of earth BEH which reata
[in the summit of the wall is here again neglected.

, Google

*2i8 EEVETEMENTS.

substituting in that equation (x—i) for c: wlienoc we obtain,
observing mat tau. aj=0, and substituting a for «„
■ _-^^— &)'tan.'a+ffl(a;— 5)tau.n+ff'_
~ {a>~b)ta.n.a+2a '

Also W=ii>.{x~i){{x-h)tB.n.a,-h2a} (4ol);

:.'WX=ili.{x—b}\^{x—b)'taxi.'a+a{x—?')ta,T>.a + a'\.

It remains, therefore, only to determine the vahie of the
term P . X. Now it is evident (Ar't. 16.) that the prodiict
P . X is equal to the sum of the moments of the proeaures
upon the eiementaiy em'facea which compose the whole sur-
face BX. But the pressure upon any such elementary sur-
face, whose distance from A is x, is evidently represented

e represented by —^x&ie,

and the sum of tl^e moments of all such elementary pressures

by S-^x^x, or when Aa; is infinitely small, by

/ '-T-xdio : therefore P . S= / -^r-^dx.
J ax ^ J ax

But diiFerentiating equation (4i8),

dV , / !. 5 , =\ii \ a^tan.'ip )

-T-=u., Jiceec. (:— (iC tan. ii + c )* < sec.?— y^- — j — rja^-

Performing the actual multiplication of the factors in the

(ar'tan. '(p+?)i

ducing we obtain

* P being a function of x, let it be represented ^f{x)\ then will j^s) repre-
sent tlie pressure upon a pordon of the surface BX terminated at tne distance
X from A, and rtsc-f-ia:) that upoQ a portion terminated at the distance !c-\-^a;\
therefore y^ic^-'i^)— A ''''l represent the pressure iipcn the small element A2
of the surface included between these two distances. But bj Taylor's theorem,

/(3! + Aii!i-A=^?a3; + ^-^V. &c.; therefore, neglecting terms in
rolving powers of Az above the first, pressure on eitiiient -. - -- Al.

, Google

^=: ii i a; (sec. "? + tan. » —2 sec. cp {x' tan. \ + c')! +

Multiplying this equation hj x, and integrating "between t.
limits b ana x,

Ksec.>+tan.'ip)(aj'— S')~|8ec. (pcot.'<p|(37'tan.'(i

+ c')^— {jnan-'ip + e")!} +c'sec.'peot.>

Kar" tan.\ + c')*-(5'tan.''p + c^*| (452).

This value of P . X being svibstitiited in ecination (450),
and the values of Wx, w, P, from equations (448] and
(451), the line of resietanee to the revetement wall will be
detennined, and thence all the conditions of its stability
may be found as before.*

The Akch.

331. Each of the strncturea, the conditions of whose sta-
bility (considered ae a system of bodies in contact), have
hitherto been discussed, whatever may have been the pres-
sures supposed to be insistent upon it, has been supposed to
rest ultimately upon a sm-ole resisting surface, the resultant
of the resistances on the different elements of which was at
once determined in magnitude and direction by the resultant
of the given insistent pressures! being equal and opposite
to that a-esultant.

The arch is a system of bodies in contact which reposes
ultimately upon hvo resisting surfaces called its abutments.
The resistances of these surfaces are in equilibrium with the

* The limita whlcli ttie author has in tMs work imposed upon himself do not
leave liim space to enter further upon the discuasiou of this case nf tha
revetement wtiU, the applleation of vfhich to the theory of fortifloation U so
direct and obyious. Tlie reader desirous of further informatiOQ is referred to
the treatise of M. Poncelet, entitled "M^moite sur la Stabilitfi dea Eerete-
menta et de leurs Fondaliona." He will there find the subject developed in all
11* practical relations, and treated with the accostomed originality and power
of iJiat illuBtrioua author. The above method of inveatlgatlon has nothing in
common with the method adopted bj M. Poiioelet eseept Coulomb'a priaeipla
of the wedge of maiimnm pressure.

■|- The weight of the structure itself ia supposed to be included among tliese

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430 THE PKINCIPLE OF LEAST RKSISTANCE.

given pressures insistent npon tlie arch (inclusive of its
weight), bnt the direction aiid amount of the resnUant pres-
enre npon each etii-face is dependent npon the unknown
resistance of the opposite anrlace; and thus the general
method applicable to the determinafioB of the nne of
resistance, and thence of the conditions of stability, in that
large class of structures which repose on a single resisting
surface, fails in the case of the arch.

332. The peinciple of le;

If there he a system of pressures in equiUhrmm, among which
are a given number of resistanGes, then is each of these a
Tmmmum, sul^ect to the conditions imposed ly the egwU-
hium of the whole.'''

Let the pressures of the system, which are not resistances,
be represented by A, and the resistances by E ; also let any
other system of pressures which may be made to replace the
pressures B and sustain A, be represented by 0.

Suppose the system B to be replaced by C ; then it is
apparent that each pressure of the system C is equal to the
pressure propagated to its point of application from the
pressures of the system A ; or it is equal to that pressure,
together with the pressure so propagated to it from the
other pr^sures of the system 0. In the former case it is
identical with one of the resistances of the system B ; in the
latter case it is greater than it. Hence, therefore, it appears
that each pressure of the system B is a mmimum, subject
to the conditions imposed by the equilibrium of the whole.

If the resultant oi the pressures applied to a body, other
than the resistances, he taken, it is evident from the above
that these resistances are the least possible so as to sustain
that resultant ; and therefore that if each resisting point be
capable of supplying its resistance in ami/ direction, then ai-e
all the resistances parallel to one another and to the result-
ant of the otlier pressures applied to the body.

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THB AEOH. 431

333. Of all the pressvrres which own he aj^Ued to the highest
voussoir of a semi-aroh, d^erent vn their amowbts and
points of application, lict aU coTisistmt with the e^mU-
ttriiim of the send-arch, that which it would sustain from
the pressure of an opposite a/nd equal sefM-O'rchisthe least.

Let EE represent the surface by which an arch rests upon

either of its al tance'i upon the

different points t tl \i, utoci, (Ait ^>il tl e least pre=(sure6
which, beng apil edto tl ose points aiecmsistent with tlie
equilibritmi ot tne arch They aie more ^ er paiallel to one
another: their leaultant is therefoie the least single piessure,
■which, being applied to the surface EB, would be sufficient
to maintain the eCLnilibrium of the arch, if the abutment were
removed.

Now, if this resultant be resolved vertically and horizon-
tally, its component in a vertical direction will evidently be
equal to the weight of the semi-arch : it is therefore gi/oen in
amount. In order that the resultant may be a minimum, its
vertical component being thus given, it is therefore necessaiy
that its horizontal component should be a minimum ; but
this horizontal component of the resistance upon the abut-
ment is evidently equal to the pressure P of the opposite
semi-arch upon its key-stone ; that pressure is theretore a
minimum ; or, if the semi-arches be equal in every respect,
it is the least pressure which, being applied to the aide of the
key-stone, woald be sufficient to support either semi-arch ;
which was to be proved.

The following pi'oof of this property may be more intelli-
gible to some readers than the preceding. It is independent
of the more general demonstration of the principle of least
resistance.*

f this work in Mr, Hann's " Treatise on the

./Google

THE AHCH.

The preseuro which an opposite semi-arch woiilcl produce
upon the side AD of the key-stone, is equal to the tendency
of that semi-arch to revolve forwards upon the inferior edges
of one or more of its vonssoirs. Now this tendency to motion
is evidently equal to the least force which would support Use
opposite semi-arch. If the arches be equal and equally
loaded, it is therefoi-e equal to the least force which would
support the semi-arch ABED.

334. Gexkeal conditions of the bt.usiljtt of an abch.*

Suppose the mass ABDO to be acted upon hy any number
of pressures, among wliich
is the pressure Q, being the
resultant of certain resist-
ances, supplied by different
points in a sarface BD ;
common to the mass and to
an immoveable obstacle
BE.

Now it is clear that un-
der these circumstances we
may vai-y tlie pressure P,
both as to its amount, di-
rection, and poiat of appli-
cation in AC, without disturbing the equilibrium, provided
only the form and direction of the line of resistance continue
to satisfy the conditions imposed by the equilibrium of tlie
system.

These have been shown (Art. 283) to be the following : —
that it no where otit the surface of the mass, except at P,
and within the space BD ; and that the resultant i>reB8ure
upon no section MN of the mass, or tlie common surface BD
ol the mass and obstacle, be inclined to the pei-pendicnlar to
that suiface, at an angle greater than the Hmitiug angle of
resistance.

Thus, varying the pressure P, we may destroy the equi-
librium, eithei;, iirst, by causing the resultant pressureto
take a direction witliout tlie limits prescribed by the resist-
ance of any section MW through which it passes, that is,
without the cone of resistance at the point where it inter-

n Bridges, vol. i. ; Memoir bj the aa-

,y Google

THIL AECB. 4a3

sectB that surface ; or, secondly, ty cansing the point Q to
fall without the surface BD, in ■which case no resistance can
be opposed to the resultant force acting in that point ; or,
thirdly, the point Q lying within the surface BD, we may
destroy tiie equilibrium by causing the line of resistance to
ci]t the surface of the mass somewhere between that point
and P.

Let ue suppose the limits of the variation of P, within
which the first two conditions are satisfied, to be known ; and
varying it, within those limits, let us consider wlmt may be
its [east and greatest values so as to satisfy the third condition.

Let P act at a given point in AC, and in a given direc-
tion. It is evident that by diminishing it under these
circumstances the line of resistance will be made continually
■to assume more nearly that direction whicli it would have
if P were entirely removed.

Provided, then, that if P were thus removed, the line of
resistance would cut the surface, — that is, provided the
force P be necessary to the equilibj-ium, — it follows that by
diminishing it we may vary the direction and curvature of'
the line of resistance, until we at length make it t&uah som&
point or other in the surface of the ma?s.

And this is the limit ; for if the diminution he carried^
further, it will cut the smface, and the equihhrium will be
destroyed. It appears, then, that under the circamstances
supposed, when P, acting at a given point and in a given
direction, is the least possible, the line of resistance touches
the interior aurfaae or i/ntrados of the mass.

In the same manner it may be shown that when it is the
greatest possible, the line of resistance touches the exterior
surface or extrados of the mass.

The direction and point of apphcation of P in AC have^
liere been supposed to be given ; but by varying this direc-
tion and point of application, the contact of the hue of
resistance with the mtrados of the arch may be made to
take place in an infinite variety of different points, and each
such variation supplies a new value of P. Among these,
therefore, it remams to seek the absolute maximum and
minimum values of tliat pressui-e^

In respect to the direction of the pressure P, or its incli-
nation to AO, it is at once apparent that the least value of
tliat pressure is obtained^ whatever be its. point of applica^
tion, when it is liorisontal.

There remain, then, two ctmditions to which P is to be
subjected, and which involve its condition of a mhiimunu

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484 THE AKCH.

The first is, that its amount shall ie such as will give to the
line of resistance a point of contact with the intvados ; tlio
eecond, that its povnt of <3m)lication in, the Jsey-atone AO
shall be such as to give it the least value which it can receive^
subject to the first condition.

335. PkACTJCAL conditions of the BTABILTTT of ah AKCIl
OF tTNCEMEN'XED STONES.

Tlie condition, however, that the reeultaiit pressure upon
the bey-atone is subject, in respect to the position of ita
point of application on the key-stone, to the condition of a
minimum, is dependent upon hypothetical qualities of the
masonry. It supposes an unyielding material for the arch-
etones, and a mathematical adjustment of their surfaces.
These have no existence in the uncemented arch. On the
striking of the eenti-es the arch invariably sinlis at the
crown, its voussoii-s there slightly opening at their lower
edges, and pressing upon one another excluBively by their
upper edges. Practically, the line of resistance then, in an
arch of unoemented stones, touches the extrados at the crown ;
so that only the first of the two conditions of the minimum
stated above actually obtains : that, namely, which gives to
the line of resistance a contact with the intrados of the
arch. This condition being assumed, all consideration of
the yielding qnality of the material of the ai-ch and its
abutments is diminated.

The form of the solid has hitherto been assumed to be
given, together with the positions of the different sections
made through it ; and the forms of its lines of resistance and
pressure, and their directions through its mass have thenco
been determined.

It is manifest tliat the converse of this operation is pos-
sible.

Having ^ven the form and position of the line of resist-
ance or of pressure, and the positions of the different sections
to be made through the mass, it may, for instance, be
inquu'ed what form these conditions impose upon the surtace
which bounds it.

Or the direction of the line of resistance or pressure and
the form of the bounding surface may be subjected to certain
conditions not absolutely determining either.

If, for instance, the form of the intrados of an areh be
given, and the direction of tlie intersecting plane be always

./Google

perpendicular to it, and if tlio line of pressure l)e supposed
to intersect this plane always at the same given angle with
the perpendicular to it, so that the tendency of the pressure
to thrust each from its place may be the same, we may
determine what, under these circumstances, must be the
extrados of the arch.

If this angle emal constantly the limiting angle of resist-
ance, the arch is in a 8tat« bordering upon motion, each
voussoir being upon the point of slipping downwards, or up-
wards, accor<Sng ns the constant angle is meaeured above or
below the perpendicular to the surface of the voussoir.

The systems of voussoirs which satisfy these two con-
ditions are the greatest and least possible.

If the constant angle be zero, the line of pressure being
every where peTOendicular to the joints of the voussoirs, the
arch would stand even if there were no friction of their sur-
faces. It is then technically said to be equilibriated ; and
the equilibrium of the arch, according to this single con-
dition, constituted the theory of the ai'ch so long in vogue,
and so well known from the works of Emei-son, Hutton, and
Whewell. It is impossible to conceive any ari'angement of
the parts of an arch by which its stability can be more
effectually secured, so far as the tmdency of Us ixmssoirs to
slide upon one n/nother is eoria&med: there is, however,
probably, no practical case in which this tendency really
affects tne equilibrium. So great is the limiting angU <w
resistance in respect to all 9ie kinds of stone used in the
construction of arches, that it would perhaps be diffiffult to
construct an arch, the resultant pressure upon any of tlie
joints of which above the springing should lie without this
angle, or which should yield by the sapping of any of its

Traced to the abutment of the arch, the line of r
ascertains the point where the direction of the resultant
pressure intersects it, and the line of pressure determines the
inclination to the vertical of that resultant;* tliese elements
determine all the conditions of the equilibrium of the abut-
ments, and therefore of tiie whole structure ; they associate
themselves directly with the conditions of the loading of the
arch, and enable us so to distribute it as to throw the points
of rupture into any given position on the intrados, and give
to the line of resistance any direction which shall best con-

* The inclination of the tesultant pressure at the springing to the vertica]
may be determined independently of the line of pressure, as will hereafWv be
shown

, Google

43o THE LINE OF RESISTANCE IK I'llE AECH.

dnce to tlie etability of the stractnre ; from known dimen-
sions, and a known loading of the arch, they determine tlic
dimensions of piera whicn will support it ; or conversely,
from known dimensions of the piers they ascertain the
dimensions and loading of the arch, which may safely be
made to span the space between them.

336. To DE-j^KMraE the LraE of i

WHOSE IMTEADOS IS A CIECI.T5, AND WHOSE LOAD IS COL-
LECTED OVEE TWO POINTS OF ITS EXTKADOS 8YMMETKIOALLY
PLACED IN ItESPECT TO THE CEOWN OF THE AECH.

Let ADBF represent any portion of such an arch, P a
u pressure applied at its extreme

■,^ ,n Tonsfioir, and X and Y the ho-

rizontal and vertical compo-
nents of any pressure borne
upon the portion DT of its ex-
trados, or of the resultant of
any number of such pressures ;
let, moreover, the co-ordinates,
irom the centre 0, of the point
of application of this pressure,
or of this resultant preesm-e, he
X and y.

Let the horizontal force P
be applied in AB at a vertical dietan-ce ^ from 0 ; also let
CT represent any plane which, passing through 0, interaecte
the ai'ch in a direction parallel to the joints of its voussoirs.
Let this plane be intersected by the resultant of the pres-
sures applied to the mass ASTD m K. These pressures are
the weight of the ma^ ASTD, the load X and Y, and the
pressure P. !Now if pressures equal and parallel to these,
but in opposite directions, were applied at R, they would of
themselves support the mass, and the whole of the enbjjicent
mass TSB might he removed without affecting the equili-
brium. (Art, 8.)ImagiQe this to be done ; call M the weight
of the mass ASTD, and A the horizontal distance of its cen-
tre of gravity from C, and let OE he represented by f, and
the angle ECS by fl, then the pej-pendiciilar distances from
C of the presBiii-es M-|-Y and P--X, imagined to be applied
to E, are p sin, 6 and p cos. I ; therefore by the condition of
the equality of moments.

./Google

r ECPTCEE IN THE ,

CM:+T)f sin. l + (P-X)fCoa.l=lI4+Tai-X!/+Pp;
Mi+Ta-Xy+Pj)

~(M+Y)sm.i! + (P-X)ooa. 1 '

■ («3),

wMch is the equation to tiie line of resistance.

M and h are given functione of ^ ; as also are X and T, if

the preeaure of the load extend contmiioush/ over the surface

of the extrados from D to T.

It remains from this equation
to determine the pressure P, be-
ing that supplied by the opposite
semi-ai'ch. As the simplest caae,
let all the vonssoirs of the arch
be of the same depth, and let the
inclination ECP of the first joint
of the semi-arch to the vertical be
represented by 0, and the radii
of the extrados and intrados by
E and r. Then, by the known
principles of statics.*

r' sin. klidr= —4(31'— '/)( cos. S— cos. 9) ;

30, M=i{Ii"-»-'Xfl-0) ;
:.p\^n'—'/'^{e—&) sin. 6+Y sin. 6—X cos. 6+V cos. fl| =

KIi'-/)(cos.0-cos.a) + Yic-X2/+Pi' (454),

which is tlie general equation to the line of resistance.

The Angle of Rtjptuke.

337. At the points of rupture the line of resistance m,eeta
the intrados, so that there p=r: if then T be the correspond-
ing value of fl,

^Jj(E'_/)(T— 9)sin. ■^'-i-Ysin. f — X cos. Y+P cos. fj =:
4(E'-/){cos. ©-cos.T) + Tie-Xj' + Pp (i55).

• See Note 1 at end of Pam IV.— Ed.

./Google

438 THE ASGLE OF HUPTUEE

Also at the points of rupture the line of resistance touoJiei

simplify the results, tliat the pressure of the load is wholly
in a vertical direction, so that X=0, and that it is collected

dY
over a single point of tlie extrados, so that -—=0, and dif-

and p=?', we obtain

T JK^ '-/) (Y-©) COS. T + J(B'-/) sill.T +
Y COS. T— P sin. f} =^{R'—r') sin. T ;
hence, assuming 31=r (1 + a),

I ^+.-{2«-f 3) } faii.T= { ^-3cc(. + 2)e I +

3a(a + 3>f (456).

Eliminating (T— ©) between equations (455) and (456), we
have

{J+^'(4<' + i)|sec.'T_|^+^+a(K+«+l)cos.©|

sec.-i'=-a(ia + l) (457).

Eliminating P between eriuationa (455) and (456), and
reducing,

V p.o..T+»rin.y_^ J =(i,. + .)(l_£c„,T)(T_e) +

£(K+i»')»in-^- !(«+«'+*•') OB- ®-8»" + '>)i!M-''!»m.'<'
' (458).

' Thia eqimtion miglit have been obtained by differentiating equation (154)
m reapEct to P and 0, and assuming -„ = 0 when r and "f are substituted for
p and 0; for if that equation be represented by i!=0, « being a function of

therefore obtained, whether we aasume -^ = 0. "'^-la =f'' "^'•i*' '^'^ suppoai-
Uon is that made in equation (456), whenoe equation (458) has resulted. Tho

, Google

IK THE AECH.

Let AP = >-r: thereforo '^s
r
(l+>-) COS. 0. Subatitnting this

value of — ,

-, I ^ sin. T + (l + X) COB. e COS. Y-^1 j = {ia'+ a)
I }1— (l+>^)co8.0 eos.'J'i(Y~©) + (cos.T— coa.©)sin.Y t ^

^(i"" +i«°) sin. 'i' COS. 0 .

. (459),

by which eqiiation the angle of rupture f is deteiTuined.

If the arch be a continuous segment the joint AD ie ver-
tically above the centre, and CD coinciding with CE, 0=0;
if it be a broken segment, as in the Gothic arch, © has a
given value detei-mined by the character of the arch. In the
pure or equilateral Gothic arch, © = 30°. Assuming 0=0,
and reducing,

^ j^-|tan.^->.cot.yj |=(V+«)| (taii.^->.cot.TJ

T-vere. Y I +X{V+K) (460.)

It may easily be shown that as f increases in this equa-
tion, T increases, and conversely; so that as the load is
increased, the points of j-upture descend. "When Y=0, or
there is no load iipon the extrados,

(tan.^-Xcot.YJ'r_Yers.Y-i-^Xcc^t^=0 (461).

bypothesea -j

r, determine the n

lum of the pressures P, ubich
will prevent the s^riii-arch ftorg

, Google

THE LINE OF KCBTSTANCE.

"When x~0, or the load is placed on the crown of tlie
arch,

tan.-^— Xcot.'t
When - — I tan. TT"'' cot. T t^O.-j- becomes infinite:

r \ 2 I '7' '

m infinite load is therefore reqni
mgle of rupture which ia det

Solved in respect to tan. — , it g

an infinite load is therefore reqnired to give that value to the
angle of rupture which ia determined by tide equation.

f/(;)-+>.(2+>.)

. (463).

No loading placed upon the arch can cause the angle of rap-
ture to exceed that determined by this equation.

The line of kebistakoe in a circulae akch whosh

VOCSBOIRS ARE EtJIIAL, AND WHOSE LOAD IB niSTmBTTTEn
OVEE niFFEBENT POISTS OF ITS I

that the pressure of the load ia
wholly vertical, and such that any
portion FT of the extradoa sustaina
the weight of a mass GFTV imme-
diately super incumbent to it, and
bounded by the straight line GY
Inclined to the horizon at the an-
gle t; let, moreover, the weight
of eadi cubical unit of the load be
equal to that of the same unit of
the material of the arch, multiplied
by the constant factor f- ; then, re-
presenting AD by 11/3, ACF by ©,
ACT by 0, and D2 by s, we have,

aGFTV=yT

./Google

SEGMENTAL AEOH.

441

butTT=MZ-(MT+VZ), and MZ^CD^-Ej-B.'S, MT=
it COS. e, YZ = DZ tan. i = E sin. 9 tan. i. Therefore MT+
VZ = Rcos. fi-l-Esm.fi tan. i=:E Jcos. fi cos. t+sin. 9 sin. ({
sec. I = E COS. [6—1) sec. t ;

.-. TV"=EjH-/3-cos.(e-0sec.(i ;
also, 3=DZ=E sin. 9 ;

0 e

.■.areaGFTV^yTV . pde=-R''f\l + ^~

e 9

COS. (fl— i) see. i\ cos. M9;

9
.-. T=weigbt of mass GFTY =(J:E'* f [1 -f ,S— sect cos.(0— ()|

6
COS. Ode = nE' I (1 + /S) (sin. fi— ain. ©)— i sec. ( jsin. (2 fl-t) —

Bin.(2 0-()}— i-(9-e)| . . . (464).*

Yx = moment of G]?Ty = i^-'R' f {{1 +/3)-sec. ( cos. [6- 1\
9
sin. e COS. ede=,j.'R' j^(l +^) (cos. °0 - COS. =6)-^ (cos. 'e —
COS. '6)—i tan. t (sin. '6— sin. '&)] . . . 466).*

A SEOJIEXTAL ARCH "WHOSE EXTRAI)03 IS HORIZONTAL.

339. As thesimplest case, let US first
suppose DT horizontal, the material of
the loading similar to tliat of the arch,
and the crown of the arch at A, so
that (=0, v=l, and ©=0. Substi-
tuting the values of T and Tic (eq^na-
tions 464, 465) which result from theso
suppositions, m equation (455), solving

that equation in respect to - •

-, and r

ducing, "^

i. of Pabt IV.— Ed.

./Google

i[l-g)(l+n)°(l+fflsia.'<?+Hl+")'(I-aa|c03.°t+[ia'-BQ'~l}coB.t-itBin.t+i
1+A-cos.* (466).

Assuming -^ =0 (see note, page 438.), and X=a, and
reducing,
|(l-2«)cos.'T_j(l_a)(l + y3) + (l + „)(l_2f.)icos.'T +

111 tiie case in wliicli the line of resistance passes throngh
the bottom of tlie key-stone, so that >-=0, equation (466)
becomes

(1 + cos. 'r)co3.'*^— i'i'cot.jT+^^O (4(38);

=0, "we have

|(l+„)=(l_2a)coe.=T + (l + „)'j(l-«)^+i(4-5t.)}cos.-i' +
^-{(l + a)'(l-^a)(l+^)+H-«Xl + *«)S=0. . . .(469.)

AECH, THE EXTKAnOS OF KAOH SEMI-AECH BETNO
LIKE IKCLJNEn AT AXT GJVEK ANGLE TO THE
HORIZON, AND THE MATERIAL OF TIIE LOAnjNG DIFFEKEMT
I'BOM THAT OF TOE ARCH.

340. Proceeding in respect to this general case of the
stability of the circular arch, by precis^y the same steps as
in the preceding simpler ease, we obtain from equation
(455),

(K+«»+<.)(oo..9-oo..^)-{io=+^)(*-6>in.l'-l-5-^siii.-l-
7=- i....^(i+.).o..e} — -(^^0,

./Google

THE GOTHIC AECH. 443

in which equation the values of T and Ya; are those deter-
mined by substituting f for & in equations (464) and (465).

Differentiating it in respect to t, assuming-^— =0 (note,

p, 438,), and >^=a, we obtain

(a+K— V— K) C03. ® sin. f— (V+a) sin. f C03. Y—

{ia+o:){l-{\ + a.) COa. 0 cos. Y}(-'i' — ©)

Y Yx

— {l-(l + o) COS. '*■ COS. ©}3-+-^ain.T+ jcos.T—

/, , > ! 1 <^(Yaj) sin. ■1'<;Y| ^ ,^„,,

{% + .) COS. ei I -, V-—- ^b'^ • ■ ■ ^*^1>

Y Ya;
Substituting in this equation the values of— and — j , de

tei-mined hj equations (464) and (465) the following eqna
tion will be obtained after a laborious reduction : it deter
mines the value of '^ :

A+B COS. T-C COS." T+D COS.' f +E sin. Y—
F sin. Y COS. Y— G sin.' Y— H cot. Y +

■ 1(1 -K COS. T)^^t-5+-^=0 (472)

^ -'sm, Y sin, i' ^ '

A=p.(l + a)' j |(l + a) tan ( sin' 0~(1 + S) }2-(l+a) cos' ®]

-|(l+a) COS.' © J +(3a+a=-cc'-K) COS. 0
E=(H-«y{2Kl-«') (1+/S) COS. ®-(l-i^)} +1.
C=Kl + «)' Kl-«) (l+^)+(l+«) (l-2«) COS. ©}.

'E=!'.{l + a.)\l—2a) tan. (=fD tan. t.

F=Kl + a)Xl— 2«) t'*^- * «<^S- ®=E(l+a) COS. 0.

G=|(J.(1 + «)X1— 2a) tan. i^D tan. *.

H=Ml+«yS2(l + ,S)-sec. t COS. (©-()! sin, 20.

I=.l_(i_[*) (! + „)'.

K=(l + a)cos. 0.

L=Kl+«)"12(l+^)~sec. ( cos. (0— ()}sin. 0.

Tables miglit readily be conatrncted from this or any of

./Google

444 AN AECH eUSTAIOTNG THE PHESBtTKB OF WATBB.

the preceding equations by aesuming a series of values of'*',
and caiculating the correBpoiiding values of ^ for each given
value of a, (, f*, ®- The tabulated results of such a series of
calculations vrould show the values of "f coi'responding to
given values of a, )3, (, f-, ©. These values of 'i' oeiug snb-
stitnted in equation (4'rO}, the corresponding valu€s of the
horizontal thrust, would he determined, and thence the polar
equation to the line of resistance (equation 454),

A CIKCULAB AEOH HATING EQUAL VOUSSOIKS t
THE rKEfiSOKE OF WATER.

' SUSTAININCt

341. Let us neyt take a case of oblique pressure on the
extrados, and let us suppose it to be
the pressure of water^ whose surface
stands at a height ^B above the sum-
mit of the key-stone. The pressure of
this water being perpendicular to the
estrados wiU everywhere have its di-
rection through the centre 0, so that its
motion about that point will vanish,
and Yas— X^=0; moreover, by the
principles of hydrostatics,* the vertical
component Y of the pressure of the water, superincumbent
to the portion AT of the extrados, will equal the weight of
that mass of water, and will be represented by the formula
(464), if we assume t=0. The hoi-izontal component Xf .of
tlie pressure of this mass of water is represented by the
formula

f>
X=iJ-E' /'|l + '3-eos.flS8in.^^fl=Kl + t)VKl+i3)(cos.0-
COS. S)—i (coa. 'e— cos. "S)] (473).

Assuming then 0=0, we have (equation 464), in
to that portion of the extrados which lies between the
and the points of rupture,
Y
-5:zzz^(t+a)')(l4-/3) sin. ■^-l sin. 2 t-Jt},

and (equation 478) -5-=fA(l+a)' 1(1 +.'^) ^^rs. t— J sin. '^f,

./Google

AN AliCU SUiTAINiNG THE PKESBUKE OF WATEK. ii^

.-.^sin. T_^c09. T=^(l+ay|{l + ,S)yers. T^iT sin.'*-}

^ ^ (4:U).

Substituting this value in equation (455), making Yx—Xy=Oj
solving tliat equation in respect to — and making ~=l+x,
■we liave
r_iia'+n-Hi+'')1'i^'"-^-i''+

li, m«tea'l ot tuppoaiug the pressure of the water to he

bome by the extrados, we suppose it to

^---rnil take effect upon the intradoa, tending to

-^5^^^^ hlow up tlie arch, and if ^ represent the

/■^=^^^ height of the water above the crown of

£^33^rfir^^ the jntrados, we shall obtain precisely

n— — ~ 2^^^^ the same expressions for X and Y as

hrrrl-^-^. _— ^ before, except that r must be substituted

' ^^ i .1 (1 +a)r, and S and T must be taken

T S

negatively; in this case, therefore, ^ sin. T— -^ cos. 'f=

— fj:j(l+,S) vers. T— |T sin. "?} ; whence, by substitution in
equation (455), and reduction,

F (ic^' + a + ^)ysi^]■Y-ja + a' + V + f^(l + g)Svers.T
rf- J.+vors.'i' '^ ' -'

€) ...

Kow by note, page 438, - , =0 : differentiating equa-
tions (4f6) and (476), therefore, and reducing, we have,
yJ tan.-|-Xcot.T j-vers. y+A>^=0 (477);

wliich equation applies to both the cases of the pessure of a
fluid upon an arch with equal voussoira ; that m which its
pressure is bome by the extrados, and tliat in which it is
borae by the intrados ; the constant A representing in the

,., ic''+i«'-f^(i+^)(l + «)' , . ,,
first case the quantity — -j- ,■ _, j-,- v,"y • ; ^^d m the

irongh tlie summit of the key-stone, X must be taken=a.

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446 EQUILlEEirM OF AN ATffiH.

If it pass along the inferior edge of tlie ke}''- stone;

>-^0. In tMs second case, tan. -^It— Bin.T| =0, therefore,

4'=0; 80 that the point of rupture is at the crown of the
arch, Tor this valne of ''f ec[uations (475) and (476) become
vanishing fractions, whose ralnee are determined by known
methods of the differential calculus to be, when the pressure
is on the extrados,

£=.-i.-+/3Kl+„y.,..{«8);
when Uie pressure is on the intrados,

^=a-y"^f* (479).

It is evident that the line of resistance thns passes through
the inferior edge of the key-stone, in that state of its etjuUi-
brium which precedes its rupture, by the asami of its crown.
The con-eeponding ecLuation to the Une of resistance is deter-

P
mined by substituting the above values of -i in eq^uation

(454"). In the case in which the pressure of the water ia
sustained by the intrados, we thns obtain, observing that

isin.fl 1 COS. ^ = — (J-)(l+/3) vers. &~p sin. i\;

„° + 2«-gf^-(ia' + a' + ^) cos, d
^^V+a+»asin.e+(«-i«'+r^)cos.«-/^(l+/a) • ■ • '■*'"^>

If for any value of 6 in this equation, less than the angle
of the semi-arch, the con-esponding value of p exceed
(1 +ay^ the line of resistance will intersect the extrados, raid
the arch will Mow v^.

The equilxbridm op an akch, the contact of whose
voussoiks is geome'rkicallt accueate.

342. The equations (459) and (456) completely determine

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447

the Yalne of P, subject to the iirat

of the two conditions stated in

Art. 333,, Tiz. that the line of re-

: sistance passing through a given

Eoint in the key-stone, detei'rained
y a given value of >-, shall have a
point of geometrical contact with
the intrados. It remains now to
determine it subject to the second
condition, viz. that its point of ap-
plication P on the key-stone shall
be such aa to give it the least va-
lue which it can receive subject to
the first condition. It is evident
that, subject to this first condition, every different value of
X will give a different value of 'V ; and that of these values of
"V that which gives the least value of P, and which con-es-
ponds to ^positive value of >- not greater than a, will be the
true angle of rupture, on the hypothesis of a mathematical
adjustment of the surfaces of toe vonesoirs to one another.
To determine this minimum value of P, in respect to the va-
riation of T dependent on the variation of >. or of ^, let it be
observed that >- does not enter into equation (456) ; let that
equation, therefore, be differentiated in respect to P and '^,

dP
and let -^ be assumed=0, and T constant, we shall thence

obtain the equation

M«+2)

■ («!)•

whence, observing that

sec. '^ { 3a{a+2) J '

we obtain by elimiiiatioii in equation (456)

sin. 2T— 3T=

.(» + 2>'

-2® .

■ (tsa.),

from wliich equation T may be detei-mined. Also by equa-
tion (481)

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448 APPLICATIONS OF THE THEORY OF THE AKCH.

^=i|3a(a + 2)coB. 'T-ct'(2tt + 3)| (483);

and by eliminating sec.T between equations (45T) and (481),
and reducing,

■J=(l+>.) COS. 0=J j 4/»(« + 2) \^ + .\ia + l) I -

<ia=+a+l)cos.0-^l|. . . (484).

The value of X given tj this equation determines the actual
direction of the line of resistance through the key-stone, on
thehypothesismade, only in the case in which it is B.vo8itme
quantity, and not gi-eater than a ; if it be negative, the hne
of resistance passes through the bottom of the key-stone, or
if it be greater tlian a, it passes through tlie top.

Such a mathematical adjustment of the smiaces of contact
of the vonseoirs as is supposed in this article is, in fact, sup-
plied by the cement of an arch. It may therefore be con-
adered to involve the theory of tlie cemented arch, the influ-
ence on the conditions of its stability of the adhesion of its
voussoirs to one another being neglected. In this settlement,
an arch is liable to disruption in some of those directions in
which this adhesion might be necessary to its stability. That
old principle, then, which assigns to it such proportions as
would cause it to stand firmly did no such adhesion exist,
wiU always retain its authonty with the judicious e

Applications of the theory of the akch,

343. It will be observed that equation (459) or (473)
determines the angle Y of rupture in terms of the load Y,
and the horizontaf distance x of its centre of gravity from
the centre 0 of the arch, its radius r, and the depth ar of its
voussoirs ; moreover, that this determination is wholly inde-
pendent of the angle of tlie arch, and is the same whether
its arc be the half or the third of a circle ; also, that if the
angle of the semi-arch be less than that given by the above
eqoation as the value of "f , there are no points of rupture,
such as they have been defined, the line of resistance passing
through the springing of the arch and cutUng the iiitradoa
there.

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THEORY OF THE AECH. 449

The value of 'V being known from tliis equation, P is
determined from equation (456), and this valne of. P being
substituted in equation (464), the Kne of reaistance is com-
pletely determined ; and assigning to 6 the value AOB
(p. 437.), the coiTeepondiug value of p ^vee us the position
of the point Q, where the line of resistance intersects the
lowest vou^oir of the arch, or the summit of the pier.
Moreover, P is evidently equal to the horizontal thrust on
the top of the pier, and the vertical pressure upon it is the
weight of the arch and load : thus all the elements are
known, which determine the conditions of the stability of a
pier or buttress (Arts. 293. and 313.) of given dimensions
sustaining the proposed arch and its loading.
. Kvery element of the theory of the arch and its abutments
is involved,' ultimately, in the solution in respect to * of
equation (459) or equation (472). Unfortunately this solu-
tion presents great analytical difflciilties. In the failure of
any direct means of solution, there are, however, various
methods by which the numerical relation of T and Y may
be aiTived at indirectly. Among them, one of the simplest
is this : —

Let it be observed that that equation is readily soluble in
respect to Y ; instead, then, of determining the value of f
for an assumed value of Y, determine conversely the value
of Y for a series of assumed values of Y, Knowmg the dis-
ti'ibution of the load Y, the values of x will be known iui
respect to tliese values of ^, and thus the values of Y may
be numerically deternuned, and may be tabulated. From,
such tables may be found, by inspection, values of Y corres-
ponding to given values of Y.

The values of t, P, and r are completely determined by
equations (482, 483, 484), and all the circumstances of the
equilibrium of tlie circular arch are thence known, on the
hypothesis, there made, of a true mathematical adjustment
of the surfaces of the voussoirs to one another ; and although
this adjustment can have no existence in practice when
the voussoirs are put together without cement, yet may it
obtain in the cemented arch. The cement, by reason of
its yielding qiialities when fresh, is made to enter into so
intimate a contact with the surfaces of the stones between
which it is intei-posed tliat it takes, when dry, in respect
to each joint (abstraction being made of its adhesive proper-
ties), the character of an exceedingly thin vouesoir^ having
" - --"- "s mathematically adjusted to those of the adjacent
0 that if we imagine, not the adhesive propertiea.

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450 APPLICATIONS OF THE

of the cement of an arch, "but only those which tend to the
more unifoiin diffusion of the pressures through ite mass, to
enter into the conditions of its equilibrinm, these equations
embrace the entire theory of the cemented ai'ch. The hypo-
thesis here made probahly includes al! that can be rehed
upon in the properties of cement as applied to large struc-
tures.

An arch may fall either hy the sinting or the rising of
its crown. In the foi-mer case, the line of resistance passing
through the top of the key-stone is made to cut the extrados
beneath the points of rupture ; in the latter, passing through
the bottom of tlie key-stone, it is made to cut the extrados
between tlie points of rupture and the crown.

In the first case the values of X, T, and P, being deter-
mined as before and substituted in equation (454), and »
being assumed = (I +a)r, the value of 6, which corresponds
to f={l+ii)r, will indicate the point at which tlie line of
resistance cuts the extrados. If this value of i be less tlian
the angle of the semi-arch, the intei-section of the line of
refiistance with the extrados will take place above the
springing, and the arch will fail.

In the second case, in which the crown ascends, let the
maieimur/i, Yulue of f be determined from equation (464),^
being assumed =»' ; if this value of p be greater than B, and
ithe corresponding value of i less than the angle of rupture,
the line of resistance will cut the extrados, the arch will
■open at the intrados, and it will fall by the descent of the
-crown.

If the load be collected over a single point of the arch,
the intersection of the hue of resistance with the extrados
•wii! take place between this point and the crown ; it is that
portion .only of the line of resistance which lies he-bween these
points which enters therefore into the discussion. How if
we refer to Art. 336., it will be apparent that in respect to
this portion of the line, the values of X and T in equatioia
(453) and i(454) are to be neglected ; the only infiuence of
these quantities being found m the value of P.

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THEOEY OF THE ARCH.

Exawiple. 1. — Let a eivculai- ai"cli of equal voussotra have
the deptli of each vonssoir equal to
jVth the diameter of its intrados, bo
that a=:'2, and let the load rest upon
. it by three points A, B, D of its

[~" , ^ =-[Jj extrados, of which A is at the crown

i B D are each distant from it 45° ;
and let it be eo disti-ibiited that fths
of it may rest upon each of the points
"3 and D, and the remaining \ upon
A ; or let it be so distributed within
60° on either side of the crown as to
produce the same effect as though it
rested upon these points.

Then assigning one half of the load
upon the crown to each semi-arch,
and calling at the horizontal distance
of the centre of gravity of the load upon either semi-arch

from 0, it may easily be calculated that - =| ein. 45 =

■5303301. Hence it appears from equation (463) that no
loading can cause the angle of rupture to exceed 65°.
Assume it to equal 60°; the amount of the load necessary to
produce this angle of rupture, when distributed as above,
will then be determined by assuming in equation (460),

f=60°, and substituting a for X, -2 for a, and -5303301 for?.

Y . Y

"We thus obtain -;='0138, Substituting this value of-:, and
'r ° T

also the given values of a and "^ in equation (457), and
observing that in this eqnation^is to be talien=l+a and

©=0,we find-j = *11833. Substituting this value of -; in

the equation (454), we have for the final equation to the line
of resistance beneath the point B

■0138 ein. fl + -1183 cos. 8 + -32 a sin. S

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APPLICATIONS OF '

If the arc of tlie arch "be a com-
plete Beniicircle, tlie value of p in this

equation correepondiiig to fl = ^ will

determine the point Q, where the
line of resistance intereecta the ahnt-
ment; this value is p=l'09r.

If the arc of the arch he the third
of a circle, the valuo of p at the
abutment is that corresponding to

9 = -; thia will he found to be r, as

it manifeatly ought to he, since the
pointa of rupture are in this caae at
the springing.

In the fli-at case the volume of the semi-arch and load is
represented by the formula

'■{»■'■ + ■•)i + J }=-85M''.

and in the second case by

>■•{»«■ + «) 5+ J j=-!i«2r'.

Tlnis, siippoeing the pier to he of the same material as the
arch, the volume of its material, which would have a weight
equal to the vertieal preaaure upon ita summit, would in the
fret case be -SSSl/, and in the second case ■2442*'°, -whilst
the horisontal preasures P would in both cases be the same,
viz. 'llSSa/; substituting these values of the vertical and
horizontal pressures on the summit of tlie pier, in equation
(3t7), and for h writing ^ a~{f—r), we have in the iii-Bt

1183V-K

1 the second case,

-11833}-'~i<*''

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THEORY OF THE ABCH. 453

where H is the gi-eatest height to which a pier, whose w'dth
is a, can be built so as to support the arch.

If ■^'— ■11832y''=0, or «='4864Tj then in either case the
pier may be hiiilt to any height whatever, without being
overthrown. In this case the breadtli of the pier will be
nearly equal to ith of the span.

The height of the pier being gwwj. (as is conunonly the
case), its breadth, so that the arch may just stand firmly
upon it, may readily be detennined. As an example, let ua
suppose the height of the pier to equal the radius of the
arch. Solving the above equations in respect to a, we shall
then obtain in the first case a =■29783", and in the second

If the span of each arch be the same, and r, and r, repre-
sent their radii respectively, then t^-=t^ sin. 60' ; supposing
then the height of the pier in the second arcli to be the same
as that in tlie first, viz, ^j, then m the second equation we
must write for H, r, sin. 60°. We shall thus obtain for a the
value ■28rj.

The piers shown by the dark lines in the preceding
figures are of such dimensions as just to be sufficient to
sustain the arches which rest upon them, and their loads,
both being of a height equal to the radius of the semicirciilar
ai-eh. It will be observed, that in both cases the load
Y='0138r', being that which corresponds to the s
angle of nipture 60°, is exceedingly small.

&ample2.—Let us next take the example of a Gothic
arch, and let us suppose, as in the last examples, that the
angle of i"uptnre is 60°, and that tt= '2 ; but let the load in
this case be imagined to be collected wholly over the

ci'own of the arch, so that - = sin. 30°. Substituting in equa-
tion (459), 30° for 0, and 60° for T, and -3 for «, and sin. 30°
for -, we shall obtain the value 'SIOIS for — =: whence by

p
equation (457) — = -2405, and this value being substituted.

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A1TLICAT10K3 C

equation (45i) givea I'UBr foi

the value of p when 0 = -. "We

^ 3

have thus all the data for deter-
mining the width of a pier of
given height which will just
support the arch. Let the
height of the pier be supposed,
as before, to equal the radius
of the intrados ; then, since the
weight of the semi-arch and its
load is ■&55dr', and the horizoii-
tal thrust -aiOSr", the width a
of the pier is found by equation
(379) to be -ildbr.
' figure represents this arch ; the square,
formed by dotted Imes over the crown, shows the dimensions
of the load of the same materials as the arch which wiU cause
the angle of the rupture to become 60° ; tlie piers are of the
required width 'il^Sr, such that when their height is equal
to AB, as shown in the figure, and the arch beai's this insist-
ent pressure, they may be on the point of overturning.

Tables of the tiieust op

SM. It is not possible, within the limits necessarily
assigned to a work like this, to enter fiu'tlior upon the dis-
cussion of tliose questions whose solution is involved in the
equations which have been given ; these can, after all, be-
come accessible to the geTieral reader, only when tables shall
be fonned from them.

Such tables have been calculated with great accuracy by
M. Garidel in respect to that case of a segmental arch* whose
loading is of the same material as the voussolrs, and the ex-
trados of each semi-ai-ch a straight line inclined at any given
angle to the horizon. These tables are printed in the Ap-
pendix (Tables 2, 3).

" The term segmental arch is used, here iind elsewhere, to distinguish that
form of the circular arch in nhieh the intrados is a contiguous segment from
that in which it is composeil of two segments struck from dilTerent centres, as
in the Gothic arcii.

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THEOET OF THE ABCH. 45c

Adopting tlie theory of Coulomb*, M. Gai-idel lias anived
i.t an equationf which becomes identical with equation (472)
in respect to that particular case of the more general condi-
tions embraced by that equation, in whicb (^=1 and 0=0.

By an ingenious method of approximation, for the details
of which the reader is referred to his work, M. Garidel has
determined the values of the angle of rapture "f, and the
P ^
' r"'

0. The results are contained in the tables which will be
found at the end of this volume.

values of Y and TiC from enuatlone (464), (465), the line of
resistance is determined by the substitntion of these values
in equation (454). The hne of resistance determines the
pom4: of mt&rseotion of the resultant pressure with the sum-
mit of pier ; the vertical and horizontal components of this
resultant pressure are moreover known, the former being the
weight of the semi-arch, and the other the horizontal thrust
on me key. All the elements necessary to the determina-
tion of the stability of the piers (Arts. 289 and 312) are
therefore known.

It will be observed that the amount of the horizontal
thrust for each foot of the width of the sof&t is determined

p
by multiplying the value of —^, shown by the tables, by the

square of the radius of the intrados in feet, and by the
weight of a cubic foot of the material

* See Mr. Hana'a Theory of Bridges, Art. IB. ; also p. 24. of the Memoir oi
; Arch by the author of this work, oonttuned iu the same volume.
(■ Tables dea Toussaes des Voutes, p. 44. Paris, isr" ■"

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i TO PART rv.

The length of an eleraentarj arc ds of tbs intrados AS subtending the angle
d6 is eipresaed by rdd ; an elementarj yolume of tie arch will therefore be
eiprnsaecl by rdddr; the perpendicular diatanoe of the centre of gravity of
this Tolvime from the vertical Une CE is r sin. 0; the moment of this volume,
with regard to CB, Is therefore rJMrXf ein.9=r'ii- sin. SJfl; then from (Art,
81.1 equation (20) there obtains

r e

Note 2. Part lY, — Generalintegrals of equations 4:64:, iSS.
The general integral, (equation 464)

J'll+S-coB. (e-i ) sec. 4 COS. edff=J\l+0) COS. e-si-

/sec. I (cos. e COS. i-|-sin. 9 sin. t) ooa, Me= /(1+S) cos. ddS~

But f (1+0) COS. B<W={l+i3) sin, fl;yiw. , coa, . cos. '0de=
eec, .COS. iy(^-4^"-^)de=sec. :cos.((^e+L sin. ie)-J',6o. t mn c

,_f\^ l+^_eoB. (e-O sec < [cos. 8d9={l+ii) sin. fl-i sec. .
(sia. 30 COS. ,-9in.i ooa. ^e)-\6= (1+^) sin. 9-jSec, < sin. {"J-i)-\e

The jeneraHntegral, f{(l+0]-sec. i cos. (e~j)l ein. fl cos. SiS, {ecjuation
46B), =/(l+,3)ain. ecofl. Mfl-Z'sec. i)eos.ecos.i+sIn.l?ein. ilain.ecoa.Me.
Bnt y(l+;3) sin. e COS. ede:=f{l+3f^±^i^C^=^^(l+ff)c'>s.iB=
-{l+g)i^-^-^) = -(l+0)^^ + \{l+3).

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NOTES TO FAET IV. 457

fsec.c\ COS. 0 cos, .+aiii. Ssin. i | sin. 6 cos. i)d . e=fcas.'e f^in. Hrf0+

_i-cos.'»+l,in.'«
.■./\ |l+8)— CO. . co,.(9-.) jsta. »co.. »i . +J»=-l(l+B)co..'lH-

i(l+S)+J-COS.'«-^t»». . .ll>.'».l.B ..

Note 3. Part IV.

In equation [427], (Art. 319), bj nmkiiig ^=0, we obtain F=i /ii i" ; since

taB. T=li unii this answers to tlie case of tlie iiorizontal pressure of a perfect

fluid lijie water. From tliis espreasloa there obtains dF=ii,3:dic, to express
tJie elementary pressure at any deptli x below the surface. This depth in
(Art. 341), e(iuation(473),LsTV=AD+AB=AD+AC-BC=flK+R-Kcoa.P,

.-. rfP=uE(14-5-eoa. e)M(I+;3-cos. e)^/iE' jl+/3-oos. S] sin. m
8
.-, F=S=MKy {1+/3-0O9. e \ Bin. ddB.

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FAR T V.

THE STRENGTH OF MATEEIALS.

345. From numerous experiments whicb have been made
upon the elongation, flexure, and torsion of solid bodies
under the action of given preasui-es, it appears that the
displacement of their pai'tides is subject to the following
laws.

lat. That when this displacement does not extend beyond
a certain distance, each pai-ticle tends to return to the place
which it before occupied in the mass, with a force exactly
proportional to the distance through which it has been
displaced.

Sdly. That if this displacement be carried beyond a
certain distance, the particle remains passively in the new
position which it has been made to take up, or passes ftnally
into some other position different from tnat from which it
was originally moved.

The effect of the first of these laws, when exhibited in
the joint tendency of the particles which compose any finite
mass to retui'n to any position in respect to the rest of the
mass, or in respect to one anotlier, from which they have
been displaced, is called elastdoHy. Tliere ia every reason to
believe mat it exists in all bodies within the limits, more or
less extensive, which are imposed by the second law stated
above.

Tlie force with which each separate particle of a body
tends to return to the position from which it has been
displaced varying as the displacement, it follows that the
force with which any aggregaidon of such pai'ticles, consti-
tuting a finite portion of the body, when extended or
compresfled withm the limits of elasticity, tends to recover
its form, that is the force necessary to teep it extended or

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ELONdATION.

459

compressed, is proportional to tlie amount of the extension
or compresBion ; so that each eqnal increment of tlie extend-
ing or compressing fores produces an eqnal increment of its
extension or compression. This law, which constitutes
perfect elasticity, and which obtains in respect to fluid and
gaseous bodies as well as solids, appeara fii-st to have been
established by the direct experiments of S. Gravesaiide on
the elongation of thin wires.*

It is, however, by its influence on the conditions of
deflexion and torsion that it is most easily recognized as
characterizing the elasticity of matter, under all its solid
forms, f within certain limits of the displacement of its
particles or eleuients, called its elastic limits.

3i6, To determine the elo'ngation or oonvpression of a har of
a giv&n section, under a given sPfain.

Let K be taken to represent the section of the har in
sc^uare inches, L its length in feet, I its elongation or com-
pression in feet under a strain of P pounds, and E the strain
or thrust in pounds which would be required to extend a
bar of the same material to double its length, or to compress

* For K desoiiplion of the apparatua of S. Grovesaade, see Illitatrationa of
Meehamas, by the Author of thia work, '2d edition, p. 80. In one of hia
eiperiments, Mc. Barlow subjected a bar of wrought icon, one square inch iu
aeotion, to strains iaoceaaiiig suooesaiyeLj from four to nine tons, and found the
elongationa corresponiliiig to the successive additional strains, each of one ton,
to be, in millioDths of the wliole length, of the biir, 120, 111), 130, 130, 130.
In a second eiperiraent, made with a bar two square inches in section, under
straina increasing from 10 tons to 30 tons, he foiind the additional elongations,
produced bj sueces^re additional straina, each of two tons, to be, in milliontlis
of the whole length, 110, 110, 110, 110, 100, 100, 100, 100, 85, 90. Froni an
exten^ve eeries of sinular results, obtained from iroa of different qualities, be
deduced the conclu^on that a bar of iron of mean quality might be assumed
to elongate by 100 millionth parts, or the 10,000th part, of its whole length,
under every adflilJonal ton str^n per aquaie inch of its section. (Bepart ta
Diraetora of Lotubm and BiTmiTujham Sailmay. Fellowea, 18S6.)

The French enpneers of the Pont des Invalides asagned 82 millionth parts
to this elongation, their esperimenls having probably been made upon iron of
inferior qufJity. M. Vieat baa assigned 91 millionth parts to the elongation
of cables of iron wire (No. 18.) under the same circumstances, MM. Minard
uid Deaormes, 1,118 milUontli parts to the elongation of bars of oak,
[mml. Meek., p, BflS.)

f The experiments of Prof. Bobison on torsion show the existence of this
property in substances where it might little be expected; in pii)e-clay, for

, Google

i60 THE "WOEK EXPIiNDED ON ELONGATION.

it to one half its length, if the elastic limit of the material
were such as to allow it to he so fai' elongated or compressed,
the law of elasticity remaining the same.*

Now, suppose the har, whose section is K square inches,
to be made up of others of the same length L, each one inch
in section ; these will evidently be K m number, and the

p
strain or the thrust upon each will be represented by w-

Moreover, each bar will be elongated or compressed, by this
strain or thmst, by I feet ; so that each foot of the length of
it (being elongated or compressed by the same quantity as
each other foot of its length) will be elongated or comprised

by a quantity represented, in feet, by y. But to elongate

or compress a foot of the length of one of these bars, by one
foot, requires {by suppositioiu E pounds strain or thrust ; to

elongate or compress it by t- feet requires, therefore, E-?

pounds. But the strain or thrust which actuallv produces

P Pi

this elongation is =^ pounds. Therefore,^ = E-p.

34T. To find th^ numher of wnits of work &^ended upon the
elongation hy a giom, qurniHty (l) of a, 'ba/r whose section is
K wnd its length L.

If X represent any elongation of the bar {x being a part
of I), then is the strain P corresponding to that elongation
KE

done in elongating the bar through tlie small additional

KE
space As;, is represented by -r-at'^x (consideiing the strain

to remain the same through the small space ^x) ; and the

" The mlue of E in respect to any materia! is called tlia wotfiiius of its alas-
tidty. The Tslue of the moduli of elasticity of the principal materials of con-
Btrnetion have been determined hj experiment, and will be found in a table ai
the end of the volume.

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THE WOHK EXPENDED ON ELONGATION. 461

whole "work U done is, on thia supposition, represented by
^p-SajAiB, or (supposing Ax to be iniinitely small) by
KE / , , ,KE

L

- / xdx or by ^

348. By equation (485) T =^l, therefore JJ — i^l;

whence it follows that tlie work of elongating the bai" is one
half that which would have been required to elongate it by
the same quantity, if the resistance opposed to its elongation
had been, throughout, the same as its extreme elongation I.
If, therefore, the whole strain P corresponding to the
elongation I had been put on at once, then, when the elonga-
tion I had been attained, twiee as much work would have
been done upon the bar as had been appended upon its
■ ■ This work would therefore have been aoeumit-

lated in the bar, and in the body producing the strain under
which it yields ; and if both had been free to move on (as,
for instance, when the sti-ain of the bar is produced \>j a
weight suspended freely from its extreinity), then would
this accumulated work have been just sufficient yet further
to elongate the bar by the same distance Z,* wliieh whole
elongation of 2Z could not have remained ; because the
strain upon the bar is only that necessary to keep it
elongated by I. The extremity of the bar would therefore,
under these circumstances, have oscillated on either side of
that point which corresponds to the elongation I.

• The meohanieal principle inTOlTed in thla result haa nnmeroua appKca-
lions ; one of these is to the effaet of a sudden sanation of the pressure on a
mercurial colvimn. The pressure of such a column varying directly with its
elevation or deprearfon, follows the same law as the elasdeity of a bar;
whence it follows that if atij- pressure be thrown at onci or instantaneously
upon the surfn.ee of the mercury, the variation of the height of the oolumn
will be twice Uiat which it would reoeiTB from an equal pressure gradually
accunmlated. Some angular errors appear to hare resulted from a neglect of
this principle in the discussion of esperimenla upon the pressure of steam,
made witli the mercurial column. No such pressure can of course be made to
Operate, in the mathematical sense of the term, instantanermdy ; and tlie 1«rm
ffradvaily has a relatire meaning. All that is meant Is, that a certain relation
must obtain between the rate of the increase of the pressure and the amplitude
of the motion, so that when the pressure no longer ii

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463 EEsrijEsoK and

349. Eliminating I between equations (485) and (486), -v
obtain

.(487);

whence it appears that the woi'k expended upon the elonga-
tion of a bar under any strain varies directly as the square
of the strain and the length of the "bar, and invei-sely as the
area of its section,*

The Moduli of eesiliekck asd pkagilitt.

350. Since U^JEIyI KL (equation 486), it is evident

that the different amounts of work which must be done upon
different bars of the same material to elongate them by equal

fractional partsJY)- ai'e to one another as iJie product 3CL.

Let now two such bars be supposed to have sustained that
fractional elongation which con-esponds to their dastieUmU;
let Us represent the work which must have been done upon
the one to bring it to this elongation, and Ms that upon the
other: and let the section of the latter bar be one square
inch and its length one foot ; then evidently

U,=M,KL (i88).

Mb is in this case called the modulus of longihidinal resili-

It is evidently a measure of that resistance which the
material of the bar opposes to a strain in the nature of an
impact, tending to elongate it beyond its elastic limits.

If M/be taken to represent the work which must be simi-
larly done upon a bar one foot long and one square inch in
section to produce fractxire, it will be a measure of that
resistance which the bai" opposes to fracture under the like
circumstances, and which resistance is opposed to its fra-

* From this formula may be determined the amount of work expended prc>
judicially upon the elaatidtj of rods used for transinitting work in machinery,
mider a redprocatjng motion — pump rods, for ioftance. &. sadden effort of tbe
pressure transmitted in the nature of ao impact may make tbe loas of work
double that represented by the formula ; the one limit bring the minimum, and
the other the maximum, of the poedbte loss,

■|' The term "modulus of resilience " appears first to hare been used by
Mr. Tredgold !q his work oq " the Strength of Cast Irou," Art. SU4.

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A BAK SUSPEMDED VEETICAILT. iHd

gility ; it may therefore be distinguished from the last men-
tioned as the modidua of fragility. If Ur represent the work
which must be done upon a bar whose section is K squai-e
inches and its length L feet to produce fracture ; tiien, aa
before,

TJ;.=M^KL (489).

If Pf and P/ represent respeetiveh' the strains which
would elongate a bar, whose length is L feet and section K
inches, to its elastic Umits and to rupture ; then, equation
(487),

■'■*'=*S- SimaarlyM/=i|^ (490).

These equations serve to determine the values of the
moduli M« and M/by experiment.*

351. The elongation of a har suspended mrticaUy^ and sus-
taining a, given strain in the direction, of ittt length, the
infium.ce of its . otun weight hei/ng taken into the acoowni.

Let ce represent any length of the bar before its elonga-
tion, Aic an element of that length, L the whole length of tiie
bar before elongation, w the weight of each foot of its
length, and K its section. Also let the length x have become
w, when the bar is elongated, nnder the strain P and i.\& own
weight The length ot the bar, below the point whose dis-
tance from the point of suspension was ai before the elonga-
tion, having then been L— », and the weight of that portion
of tiie bar remaining unchanged by its elongation, it is atiU
represented by (L— a?) w. Now tins weight, increased by P,
constitntes the strain upon the element Aa; ; its elongation
under this strain is therefore represented (equation 485) by

* The eiperimentiS required to this determination, in respoet to tlie princi-
pal materials of conEtructiou, hare been made, and are to be found in tlie
pnblislied papers of Mr. Hodgltinson and Mr. Barlow. A. table of tiie moduli
of resJlienoe and fragility, collected from these valuable data, ia a deaidBratun;
in practical science.

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THE TEETICAL C

elongated, by ^x-\ ~tjt! — —^x, "wlience dividing by Arc.

and paeaing to tbe limit, wo obtain

^,_ F + (L-iB)w ,

(&~^+ Kl '•*^^'-

Intoijrating between the limits 0 and L, and representing

by L, tiie length of the elongated rod,

L,= (i + |j)l+j|j,L-. («2).

If the strain be converted into a thrust, P mnat be made
to assume the negative sign ; and if this thrust equal one
half tlie weight of the bar, there will be no elongation at all.

352. TuE YEKTICAL OSCILLiTIONS OF AN ELASTIC EOD OK
COKD STTSTAmiNG- A aiVKN -WEIOnT STTSPENDED i'BOM rrS
EXTEEMTl'Y.

Let A represent the point of suspension of the rod {Jig. 1,
on the next page), L its length AB before its elongation, and
^l the elongation produced in it by a given weight "W sus-
pended from its extremity, and C the corresponding position
of tlie extremity of the rod.

Let the rod be conceived to be elongated through an
additional distance GD=o by the application of any other
given strain, and then allowed to oscillate freely, carrying
with it the weight "W; and let P be any position of its
extremity dm-lng any one of the oscillations which it will
thus be made to perform. If, then, CP be represented by ce,
the con'esponding elongation BP of the rod will be repre-
sented by i?+iB, and the strain which would retain it perma-
nently at this elongation (equation 485) by -v-(^^4-a!); the

unbalanced pressure or moving foro6 (Art. 92.) upon the
weight W, at the period of this elongation, will therefore be

represented by -j^(^?4-£c)— W, or by -p ic ; since W, being

the strain which would retain the rod at the elongation i^, is

1 by -j-¥' (equation 485).

* Whevreira Analytical Statica, p. IIS.

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465

The unbalanced pressure, or moving force, upon t,lie masa
"W varies, therefore, as the distance x of tlie point P from the
given point 0 ; whence it follows by the general principle
established in Ai-t 97., that the oBcillations of the point P
extend to ecLnal distances on either side of the point 0, as a
centre, and are performed isoohronously, the time T of each
oscillation being represented by the formula

T-l

Tlie distance fi'om A of the centre C, about which the
oscillationa of tlie point P take place, is represented by
L+i?; so that, representing this distance by L„ and substi-
tuting for ^ its value, we have

. . (494).

353. Let us now suppose that when in
making its first oscillation about 0
{fig. 3.) the weight W has attained its
highest position d„ and is therefore, for
an instant, at rest in that position, a
second weight to is added to it ; a second-
series of oscillations will then be com-
menced about a new centre 0,, whose
distance L, from A is evidently repre-
sented by the foi'mnla

i.=i+2:gfi,...(t95).

So that the distance 00, of the two centres

•uiL ,

and the

greatest distance 0,D„ beneath the centre 0,, attained in the
second oscillation, equal to the distance, C^d, at which the
oscillation commenced above that point. Now C,I>, —

0,(?.=0(?, + CC,=OD + CO,=o+~; tlie amplitude (^,D, of

the S'

i oscillation is therefore 2

(-S

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466 THE OSCILLATIONS OF A LOADED EAR.

Let the weight w he conceived to be removed when tlie
lowest point I>, of tiie second oscillation is attained, a tltird
series of oscillations will then be commenced, the position of
whose centre being determined by equation (494), is identical
with tiiat of the centre C, ahont which the first oscillation
was performed. In its third oscillation the extremity of the
rod will therefore ascend to a point d, as far above the point
0 as D, is below it ; so that the amplitude of this third oscil-
lation is represented by 2CDi, or by 2C,Di + 0C„ or by
3 L+^^y "When the highest point d, of this third oscil-
lation Is attained, let the weight m>, be again added ; a foui-th
pscillfttion will then be commenced, tiie position of whose
centre will be determined by equation (495,) and will there-
fore be identical with the centre C„ about which the second
oscillation was performed ; so that the greatest distance C,D,
beneath that point attained in this fourth oscillation will be
ecjual to C^d„ or to CO, + CD, ; and its amplitude will be

represented by 2 {o-|-^^^j. And if the weight w be thns

conceived to be added continually, when the highest point
of each oscillation is attained, and taken off at the lowest
point, it is evident that the amplitudes of these oscillations
will thus continually increase in an arithmetical series ; so
that the amplitude "A„ of the w* oscillation will be repre-
sented by tl^e foi'mula

= 2\<j+{n

' KE I

The ascending oscillations of the series being made abotit
the centre 0, and the descending oscillations about 0„
if n be an even number, the centre of the n"' oscillation is
■0, ; the elongation c„ of the rod corresponding to the lowest
point of this oscillation is therefore equal to BG,+^A„; or
substituting for BOj its value given by equation (4:95), and
for A^ its value from equation (496),

_(W+m£)t

. (497).

KE
Thus it is apparent that by the long continued and

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DEFLTLXION. 467

periodical addition and siihtraction of a weight w, so small
aa to prodnee but a slight elongation or eonti'action of the
rod when first added or removed from it, an elongation <;_
may eventually be produced, bo great as to pass limits of its
elasticity, or even to break it. Numerous observations have
verified this fact: the chains of suspension bridges have
teen broken by the measured tread of soldiers ;* and M.
Savart has shown, that by fixing an elastic rod at its centre,
and drawing the wetted finger along it at measured inter-
vals, it may, by the strain resulting trom the shght friction
received thus periodically upon its surfaee, be made with
gi-eat ease to receive an oscillatory movement of sufficient
amplitude to be measured-f M. Poncelet has compared the
measurement of M. Savart with theoretical deductions
analogous to tliose of tie preceding article, and has shown
their accordance with it.

Deflkxion.

85i. The netct^al surface of a deflected beam.

One surface of a beam becoming, when deflected, convex,
and the other concave, it is evident that the material form-
ing that side of the beam which is bounded by the one
surface is, in the act of flexure, ewtended, and that of the
other compressed. The surface which separates these two
portions of the material being that where its extension ter-
minates and its compression begins, and which sustains,
therefore, neither extension nor compression, is called the

MBUTEAL SUKFACE.

355. The PosrrioN of the neittbaij a
Let ABCD be taken to represent any thin lamina of the

• Such was tbe fate of the suspension bridge at Broughton near Manchester,
the drcumetancea of which have been ably detailed by Mr. E. Hodgkinson in
the fourth volume of the Manchester Philoaophual IVanmetions. M. Navier
has ahown, in his treatise on the theory of suspension bridges (Sar les Fonts
Bnapetidas, Faria, 18S3J, that the duration of the osdUations of the chains of
a aospension bridge may in certain caees extend to nearly mi seconds ; there
might easily, in such cases, arise that isochronisra at each interval, or after
any number of intervals, betneen the marching step of tbe troops and the
oscillations of the bridge, whence would result a continually increasing don-
ation of the suspending chains.

i Mccaaique Iiidiiatrielle, p. 431, Art. S31.— Ed.

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THE iiECTKAL SL'KFACB

beam contained by planes
parallel to Hie piano of its
deflexion, and P„ P„ P, ^e
roBultants of all the pres-
sures applied to it ; aci that
portion of the neutral sur-
face of the beam which is
contained within this la-
mina, and may be called its
neutral line; PT and QV
planes exceedingly near to
one another, and perpen-
dicular to the neutral line at the points where they intereect
it; and 0 the intersection of PT and QV when produced,

Now let it be observed that the portion APTD of the
beam ia held in ecjuilibrinm by the resultant pressure P„
and by the elastic forces called into operation upon tlie sur-
face JPT ; of which elastic forces those acting in PR {where
the material of the beam is extended) tend to bring the
points to which they are severally applied nearer to the
plane SQ, and those acting in KT (wnere the material is
compressed), to carry their several points of application
farther fi'om the plane SV.

Let aR=x, SR='iai, and imagine the lamina PQVT to be
made up of fibres parallel to oR ; then will Aa; represent
the length of each of these fibres before the deflexion of the
beam, since the length of the neutral fibre SR has remained
unaltered by the deflexion. Let Sx represent the quantity
by which the fibre pq has been elongated by the defiexion
of the beam, then is the actual length Of that fibre repre-
sented by ^x-i-6x. Whence it follows (equation 485), that
the pressure which must have operated to produce this

elongation is represented by 'E—^k, &k being taken to repre-
sent the section of the fibre, or an exceedingly small element
of the section PT of the lamina. Now PT and QV being
normals to SR, the point 0 in which tbey meet, when
]>roduced, is the centi'e of curvature to the neutral line in
R. Let the radius of curvature OR be represented by R,

and the distance R^ by f. By similar triangles, 7yf>=

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— , Substituting this value of— in the expression for the

pressure which muet have operated to produce the elonga-
tion qf the fibre pq, and representing that pressure by aP,
we have

aP=^aS (498).

If, therefore, EP be represented by k, and E.T by ifcj, then
the sum of tbe elastic forces developed by the extension of

the fibres inKPQS ie represented by^^upAj; and, similarly,

the sum of those developed by the compression of the fibres

E *■
in ETVS is represented by^^opA^. Kowlet it be observed

that (since the pressures appUed to APTD, and in equili-
brium, are the forces of extension and compression acting
in BP and ET respectively, and the pressure P,), if the
pressure P, be resolved in a direction perpendicular to the
plane PT, or parallel to the tangent to the neutral line in E,
this resolved pressure will be equal (Art. 16.) to the differ-
ence of the sums of the forces of extension and compression
applied (in directions perpendicular to that plane, but oppo-
site to one ano^er) to the portions KP and ET of it respec-
tively. Representing, therefore, by 6 the inclination "ReP,
of the dii'ection of P, to the normal to the neutral line in E,
we have

E *i E *>

P, em. 6=~So?'^7c—^s„pAk.

But if k be taken to represent the whole section PT, and h
tlie distance of the point E from its centre of gravity, then
(Art. 18.)

M=2pAS— 2pA^; .•.P,sin.(l=

R ■

" EJc =

• (499) ;

which expression represents the distance of the neutral line
from the centre of gravity of any section PT of the lamina,
that distance being measured towards the extended or the
compressed side of the lamina according as fl is positive oi

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4T0 KADiirs

negative; so that the neutral Hne passes from one side to
the other of the hne joining the centres of gravity of the
cross sections of the lamina, at the point where ^=0, or at
the point where the normal to the neutral line is pai'allel to
the direction of P,.

356. Case of a recta/nguliw ieam.

If the form of the heam be such that it may be divided
into laminte parallel to ABCD of similar forms and equal
dimensions, and if the presaiire Pj applied to each lamina
may be conceived to be the same ; or if ite section be a rec-
tangle, and the preasm-es applied to it be applied (as they
tiBually are) uniformly across its width, then will the distance
k of me neutral line of each lamina from the centime of gra-
vity of any cross section of that lamina, such as PT, be the
same, in respect to corresponding points of all the laminae,
whatever may be the deflection of the beam ; so that in this
case the neutral sni-face is always a cylindrical surface.

367. Case m which the defecting pressure P, is nearly jper-
pendicula/r to the length of the heam.

In this case A, and therefore sin. a, is exceeding small, so
long as the deflexion is small at every point E of me neutral
hne ; so that h is exceedingly small, and the neub-al line
of the lamina passes very nearlvjor accurately, through the
centre of gravity of its section FT.

!58. inii RADIUS op CUKVA'roitE OF THE NEDTEAL
OF A BEAM.

Since the pressures applied to the portion APTD of the

lamina ABCD are in equilibrium, the principle of the equalitj
of moments must obtain in respect to them ; taking, there

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EADIUB OF OUBVAU'CEE. ill

fore, tlie point R, where the neutral axis of the lamina inter-
sects PT, as the point from which the raomeats are measnred,
and ohserving that the elastic pressnves developed by the
extension of Uie material in Er and itfi compression in El
botii tend to turn the mass APTD in the same direction
about the point R, and that each such pressure upon an
element aJc of the section FT is represented (equation 498} by
E

^ ^„,^ ,, .u^^.,^ v^„. ^^ ,.„,.t of the momenta
about the point R of all these elastic pressures upon FT is

the moment of inertia of PT "bout R, Observing, moreover,
that if p represent the length of the perpendicular let fall
fi-om R upon the du'ection of any pressure P applied to the
portion APTD of the beam, Pjj will represent its moment,
and spp will represent the sum of the momenta of all the
similai' pressures applied to that portion of the beam ;
we have by the pnnciple of the eq^uality of moments,

.•.i = 5S....(»0).

35&. The neuti-al surface of the beam is a cylindrical sur-
face, whatever may be its deflection or the direction of its
deflecting pressure, provided that its section is a rectangle
(Art. 356.) ; or whatever may be its section, provided that its
deflection be small, the direction of the deflecting pre^ure
nearly perpendicular to its length, and its fonn before de-
flexion symmetrical in respect to a plane pei-pendicnlar to the
plane of deflexion. In every such case, therefore, the neutral
lines of all the laminse similar to ABOD, into which the
beam may be divided, will have ec[ual radii of curvature at
points similar to E lying in the same right line perpendicular
to the plane of deflection ; taking, therefore, equations simi-
lar to the above in respect to ^1 the laminse, multiplying
both sides of each by I, adding them together, and ooserv-

ing that It and E are the same in all, we have — = ^2

In this case, therefore, I may be taken in equation (&00) to

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represent the moment of inertia of the wJiole seoUon of the
beam, and P the pressure applied across its -vvhole width.

360. The radius ofourvature of abeam whose dejkadon ts
small, cmd the mreoUon of the defieclmg presswrea nearly
perpendicular to the length of the iea/m.

In this case the neutral line is very nearly a straight line,
perpendicular to the directions of die deflecting pi'essures ;
BO that, representing its length hy a?, we have, in this case,
P=ie\ and equation (500) becomes

which relation obtains, whatever may be the foi'm of the
transverse section of the beam, I repreaentins its moment of
inertia in respect to an axis passing tlirou^ its centre of
gravity and perpendicular to the plane of deflexion.

361. The moment of inertia I of the transverse section of a.
liea/m about the centre of gravity of the section.

In treating of the momenta of inertia of bodies of different
geometrical forms in a preceding pai't of this work (Art. 82,
&c.), we have considered them as solids ; whereas the mo-
ment of inertia I of the section of a beam which enters into
equation (500) and determines the cm'vature of the beam
when deflected, is that of the geometrical area of the section.
Knowing, however, the moment of inertia of a solid about
any axis, whose section perpendiculai" to that axis is of a
given geometrical form, we can evidently determine the
moment of the area of that section about ttie same axis, by
supposing the solid in the first place to become an exceed-
ingly thin lamina {i. e. by mating that dimension of the
Borid which is parallel to the axis exceedingly small in the
expression for the moment of inertia), and then dividing
the resulting expression by the exceedingly small thickness
of this lamina. We shall thus obtain the following values
ofl:-

,y Google

363. Tor a boam witli a rectangular seotia/i,
whose breadth is represented by h and its deptli [-L^f'^hc',
by 0 (equation 61),

363. For a beam with a triangular 1 i^

section, whose base is 5 and ita height c j- 1= "gg Hi'-h^o'').
(equation 63), j

364. For a beam or column with a circular) t_i^„i
section, whose radius is o (equation 66), f — -j - .

365. To detennine the moment of inertia I in respect to a

^ beam whose transverse section is of the

' — iTi f — ' form represented in the accompanying

Hgure, about an axis al passing through

° — |— — 1( 5{g centre of gravity ; Ifet the breadth of

""" T "* the rectangle AB be represented by 6, and

I its depth by d„ and let h, and d, be aimi-

j "I lai'ly taken in respect to the rectangle EI",

' and 5, and d^ in respect to OD ; also let I,
represent the moment of inertia of the section about the axis
cd passing through the centve of CD, A„ A„ A,, the aa-eas
of the rectangles respectively, and A the area of the whole
section.

Wow the moments of inertia of the several rectangles,
about axes passing through their centres of gi-avity, are
represented by ^^h^d', A^XS -h^t^it ^"^^ ^e distances of
these axes &om the axis cd are respectively -iidi-^d^,
i((?,+(?,), 0. Therefore (equation 58),

\-i^hA' + W, + d^' A, + hh^: -^-lid, + d:f A, -j- ^b,d; ;
but A,=i,(?j, A,=M« A,=Kd, ;

.■.l,r=^iA.d.'+A,d,' + AAl+i{d, + d,yA,+l{d, + d;fA,.
Also if h represent the distance between the axes ah and od,
then (Art. 18) hA=i{d^ + d,)A,—i{d, + d,)A„ and (equation
68) 1=1,-^' A.

\I=j\{AA' + K^' + Kd,')+i:\id. + d,)'A,^-{d, + d,)'A,}-
^{{d^ + d,)A,~{d,+d,)Ar ^502i.

If d^ and d^ be exceedingly small as compared with d,^

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474 CEFLEZION OP A BEAM.

neglecting their values in the two last terms of tlie eqiiation
and reducing, we ohtain

. (503).

Tf the areas AB and EF he equal in eyery respect,
l=^{d;' + S{d, + d,)'\A, + -j\A,d; (504).

i66. TilK WOKK EXPKNDED TJPON THE DEFLEXION OF A BEAM
TO WHICH GIVEN" PEESSUEES AEE API'LIED.

If aP represent the pressure which must have operated
to produce the elongation or
compression which the ele-
mentary fibre pq receives,
by reason of tne deflexion
of the beam, Afc the length
of that tibre before the de-
flexion of the beam, and a^
its section; then the work
which must have been done
upon it, tJius to 'elongate
or compr^s it, is repre-
sented, equation (487) hy

Ep
But (equation 498) AP=^Ai, The work

(aF)' ■ Ax
* E.Ah ■

pended upon the extension or compression of pq is there-
fore represented by

,E.A3;, ^ „

And the same being true of the work expended on the
compre^ion oi- extension of everv other fibre composing the
elementary solid YTPQ, it follows that the whole work
expended upon the deflexion of that element of the beam

is represented by ^-™- Sp'Aj, or by i^^'^o^ ; for Sp'Aj repre-
sents the moment of inertia I of the section PT, about an
axis perpendicidar to the plane of ABCD, and passing through
the point P. If, therefore, a^ be taken to represent the
lengUi of that portion of the beam which lies between D

./Google

and M before its deflexion, and tlierefore tlie length of the
portion ac of its neutral line after deflexion, then the whole
work expended upon the deflexion of the part AM of the

bea/m is represented by ^ E2 r^^d-x. Bat (equation 500) ^=

F'p'

-=j^ ; whence, by substitntion, the above expression

becomes J-ir oT"^^- Passing to tbe limit, and represent-
ing the work expended upon the deflexion of the part AM
of the beam by «„

P' /'"'p'
Mi=3|, 7 Y*fe .... (505).

367. The work mp&nded wpon the deflexion of a beam of
■tmiform dmiensiona, whem, the dieting pressures aire
nearby perpendioulo/r to the surface of the ieam.

In this case I is constant, and^j=iB; whence we obtain
by integrating; (equation 605) be-
'f tween the limits 0 and a„

-■- «,=s^....(S06),

where m, represents the work ex-
pended upon the deflexion of the
i portion AM of the beam. Simi-

larly, if he=a^, the work expended
upon the deflexion of the portion BM of the beam is repre-
sented by

v:a:

BO that the whole work Uj expended upon the deflexion of
the beam ie represented by

_ PjV+P>='
^ -^ 6EI

But by the principle of the equality of moments, if a
repvesent the whole length of the beam,

./Google

INFLEXION OP A BEAM.

Eliminating P, and P, between tlieso equations and tKe pre-
ceding, we obtain by reduction

{a,a^fP'

. (607).

If the pressure Pj be applied in the centre of the beam,

. (508).

368. The likisae deflexion op a beam whes the dieection

of the deflecting- peessube is peefendicitlae to tts

ei!SFACE.

Let the section MK remain fixed, the deflexion taking
place on either side of that section ;
■ ^ then w, representing the -work ex-
pended upon the deflexion of the
portion aM of the beam, and D,
the deflexion of the point to which
J P^ is appUed, measured in a direc-
tion pei'pendicular to the surface, we

have (equation iO), u,= /p,(fD,;

, , _ du. du, (ZP *
theretore r, = ^^7 = ~rn ■ • tt.
' i^D, dF^ dJ), .

But by eqization (506), -^p- =|-^j=rj^; therefore P^^^^--^^^.
-T7y ; therefore jp- = -J tj— ; -wheuco we obtain by integration

D,:

'3EI ■

If tliG -whole work of deflecting the beam be done by the
pressure P^, the points of application of P and P, having no
motions in the directions of these pressures {Art, 52.), then
proceeding in respect to equation (507) precisely as before in
respect to equation (506), and representing the t

» Church's DIff. C'jI.

./Google

DEFLEXION OF A. BKAJT. 477

perpendicular to the surface of fhe beam at the point of
applicaiion of P^ by D„ we shall obtain

-.^^m^ (-^-

If the pressure P, he applied at the centre of the beam

■■■'^■=^ (=")•

Eliminating Pj between equations (500) and (509), and P,
between equations (507) and (510), we obtain

by which equations the work expeiided upon the deflexion
of a beam is determined in terms of the defl&eion itself, as
by equations (506) and (507) it was determined in terms of
the deflecting pressures.

169. CoKDmONS OF THE DEFLEXION OF A BEAM TO WHIOH ARE
APPLIED THREE PBESeUKES, WHOSE DIRECTIONS ARE NBAKLT
PERPENDICULAH TO ITS SUKFACE.

Let AB represent any lamina of the beam parallel to its

plane of deflexion, and aeb the neutral line of that lamina
intersected by the direction of P, in the point c.

Draw ccx, parallel to the length of tlie beam before its
deflexion, and take this line as the axis of the abseissse, and
the point 0 ae the origin ; then, representing by a; and y the

./Google

THE NEU'l-RAL LINE.

co-ordinates of any point iu ao, and by E tiie radius of cnrva ■
ture of that point, we have*

Xow the deflexion of the beam being supposed exceed-
ingly email, the inclination to ea; of 3ie tangent to the

neutral line is, at all points, exceediugly small, so that

may

m]

If~(fo^'

Substituting this value in equation (501), and observing that
in this case^ is represented by (a,— ai) instead of a;,

~ EI ■

, (513).

the direction of the pressure P, being supposed nearly per
pendicular to the surface of the beam, ana I constant. Let
the above equation be integrated betweeix the limits 0 and
«, |S being taken to represent the inclination of tlie tangent

at 0 t« i3x, so that the value of -^ at (j may bo represented by

-EI''

Integrating a second time between the limits 0 and a;, and
y that when ai^O, y=0,

_ g similarly in respect to the portion Sc of the neu-

ti-al line, but obsei-ving that in respect to this curve the value

of —^ at the point c is represented by tan. /3, i

dd'~ EI '

" Churcli's Diff. Ca.1. -Irt. ]

./Google

EQCiTION TO THE NEUTEAL LINE. 479

j/=||{i«^-K(-«.taii.3 . . . (517.)

If D, and D, be taken to represent the deflexions at the
points a and i, and ea and ch be asBumed respectively equal
to cd and oe,

Pa'
by equation (517), Di~~~—a^ tan. fl.

If the presBures P, and P, be supplied by the resistances
of fixed surfaces, then Dj^^D,. Subtracting the above eijua-
tion we obtain, on this supposition,

3E1

— = +(», + «,) tan. /^.

Now F,a,'— Pa'= '°''°' ■ ^^^=p3'*,«!j(«i— «i)i ob-
serving that P,([=Pja„ Pj«=Ps«i, andi2,+a5=(j,
/. tan. /3^^^^^ . . . (518).

If ,8„ /3, represent the inclinations of the neutral line to
iWBi at the points a and h, then by equations (614) and (516)

___ _P^,"

-2EI' '•'"'■'-'-^■^^■' 21^1-

Substituting for tan. ^ its value from equation (518), elimi-
nating and reducing,

To determine the point m where the tangent to tlie neutral
line is parallel to «ca!„ or to the undeflected position of the

beam, we must assume y^=0 in equation (518)*; if we

then substitute for tan. /3 its value from equation (518),
substitute for Pj its value in terms of P,, and solve the

* Church's Diff. Cd. Art. T8.

./Google

iSO LENGTH OF THE NEUTRAL LIKE.

resulting equation in respect to x, wo aball obtain for the
distance of the point m from o the expression

*,+ Viala^+'Za^ (530).

370. The lesgti: of the nehteal lijse, t
loaded rs the csntee.

Let the directions of the resistances upon the extremitiea

of the beam he supposed nearly perpendicular to its surface ;
then if ic and y be the co-ordinates of the neutral line from
the point a, we liave (equation 601), representing the hori-
zontal distance AB by 2a, and observing that in this case

Integrating between the limits x and a, and observing that
at the latter limit -j- = 0,

Now if s represent the length of the curve ao,

■ Ohuroh's lut. Cal. Act. IBt.

./Google

THE DEFLEXION OF A BEAM.

the deflexion being ainail, -57' is exceedingly small at every
point of the nentral line.

•••»=«+(ISp----(™)-

Eliminating P betweeTi this eqiiation and oq^nation (511), and
representing the deflexion hy B,

iTl. A BEAM, ORE PORTIOIT OF WHICH IS ElBMLT IN8SHTBD IN
MASONRY, AND WHICH SITSTAINS A LOAB UNIFORMLY DISTEI-
JiUTEU OVER IT9 REMAIHISG POKTrON.

Let the co-ordinates of the neutral line be nif asured tiomi

• The foiloning eiperimenfa were made by Mr. Hatolier Bupcimtendant of
the work-shop at King's College, to varify this result, which la identical with
that obtained by M. Ndvier (Sesuwie rfea i«fonj, Avt. 86 ) Wrought iron
rollers -7 inch in diameter were placed loosely on wrought iron bare, the aur-
fiicea of contact being smoothed with the file and well oiled The bar to be
tested had a square section, whose side was '1 inch, and was supported on the
two rollers, whiob were adjusted to 10 feet apart (centre to centre) when the
deflecting weight had been put on the bar. On removing the weights care-
fully, the diattinee to which the roUera receded as the bar recoTeced its hori-
Kontal position was noted.

-•■IS"-'

,.....,„,.„,-.

Blslance through Bhicl

DIstauee-UiTDi^li nhidi
Piwh Rotbi- would have
recefledbyPomBla.

56
84

5-*5

■2

■18

, Google

THE DEFLEXION OF A BEAM

the point B whei'e the beam
is inserted in tlie masonry,
and let the length of the
portion AD which sustains
the load be represented by
a, and the load upon each
niiit of its length by (J. ;
then, representing by a> and
1/ the co-ordinates of any
point P of the neutral line,
and obeerviiig that the pres-
sures applied to AP, and in
eq^uilibrium, are the load ii-(a,—x) and the elastic forces
developed upon the transverse section at P, we have by the
principle of the equality of moments, taking P as the point
from which the moments are measured, and observing that
ance the load i'-{a—x) is uniformly distributed over AP it
produces the same effect as though it were collected over the
centre of that line, or at distance i{a—x) from P ; observing,
moreover, that the sum of the moments of the elastic forces
upon the section at P, about that point, is represented {Art.

868.) by J|, or by EI § (Art. 369.) ;

EI^=«»-»)'--.(522).

Integrating twice between the limits 0 and is, and observing
that when ai=0,-^=0 and y=0, since the portion BO of the
ibeam is rigid, we obtain

Elg= -i,,.{a-xy+li>.a' . . . {623),

Kl2/=j'i)Ji{ffi— i»y+>a'i»— sVl^a* ■ ■ ■ - {524),

■which 18 the equation to the neutral line.

Let, now, a be substituted for x in the above equation ;
and let it be observed that the corresponding value of y
represents the deflexion D at the extremity A of the beam ;
■we shall thus -obtain by reduction

-8E1'

. {525).

./Google

LOADED TNIFOKMLT.

Representing by 0 tbo inclination to the horizon of the tan-
gent to the ncuti-al line at A, substituting a for x in eciuation

(523), and obeerving that when x=a, -^= tan. ^, we obtain

taa.;S=

. (526).

372. A BEAM 8"CPP0KTED AT ITS EXTREMITIES AND SUBTAINrNG
A LOAD UNIFORMLY DISTEtBUTSD OVUR ITS LENaTH.

Let the length of the beam be represented by ^a, the load
upon each unit of length by f^ ; take
X and y as the co-ordinate of any
point P of the neutral line, from tlie
origin A; and let it be obserred
that the forces applied to AP, and in
equilibrium, are the load fj* upon that
^ portion of the beam, which may be
^"^ supposed collected over its middle
point, the resistance upon the point A, which is represented
by (ia, and the elastic forces developed upon the section
atP; then by Art. 360.,

=iiu^-M^ (527).

Integrating this equation between the limits x and a, and
observing that at the latter limit ^ = 0, since y evidently
1 value at the middle C of tlie beam,

dx

= ilJ-{x'—(^—im(a^—a') .

. (528).

Integrating a second time between the limits 0 and x, and
; that wlien a!=0, y=0.

Elt/=ii^{ix'—a'x)-ifi.a(^x'—a'x) (529),

which is the equation to the neutral line. Substituting a for

./Google

4S4 TIIE DEFLEXION OF A BEAM

ft! in this equation, and observing that the corresponding
7alue of y represents the deflexion D in the centre of the
beam, we have bj reduction

D=^....(530).

Eepreeenting by ^ the inclination to the horizon of the tan-
gent to the neutral line at A or B, and observing tliat when

. (631).

Lot it be observed that the length of the beam, which in
equation (611) is represented by «, is here represented by
2», and that equation (530) may be placed under the form

D=|-. ■■■■■■^4;v ; whence it is apparent that ^e deflexion

of a beam, when uniformly loaded throughout, is the same
as though f ths of that load (3ofj.) were suspended from its
middle point.

373. A BEAM IS SUPPOKTED BT TWO STBTJTS

MBTKIOALLT, AND IT IS LOADED UNIFORMLY THROUGnOUT
rrS WHOLE LENGTH ; TO DETERMINE ITS DEFLEXION.

Let CD=;2a, 0A=»„ load upon each foot of the length

■ , , . . . of the beam=n; then load on

i^^'L^" """--'' -^-'- ~^ each point of snpport=[Aa, Take
^^^j . "" — - -~ -" -^ C as tlie origin ot the co-ordinates ;
'"■"' 'iji I. "~; 1 c then, observing that the forces
impressed upon any portion CP
of the beam, terminating between
C and A, are the elastic forces
j-jj upon the transverse section of the
beam at P, and the weight of the
load upon CP ; and observing that the weight c-CP of the
load upon CP, produces the same eff'ect as though it were
collected over the centi-e of that portion of the beam, so that
its moment about the point P is represented by p', OP.iOP,

./Google

LOADED nUIFORHLT. 48i>

or by ^CP''; we obtain for the equation to the neutral line
in respect to the part OA of the beam (Art. ?M)

^^ S=^^^' ^°^^>

Since, moreover, the forces impressed upon any portion CQ
of the beam, terminating between A and E, are the elastic
forces developed upon the transverse section at Q, the
resistance i>-a of tlie support at A, and tlie load upon CQ,
whose moment about Q is represented by 4ij-CQ^ we have
(equation 601), representing CQ by x,

El'^=if^ic'-K»'-«0 ..... (533).

Kepresenting the inclination to the horizon of the tangent to
the neutral Ime at A by /3, dividing equation (532) by (*,
integrating it between the limits x and a,, and observing

that at the latter limit 3^=tan. 0, we have, in respect to the

portion CA of the beam,

Integrating equation (533) between the hmits x and a, and
obsei'ving that at the latter limit -^=0, since the neutral
line at E is parallel to the horizon,

7 2=*"'-W"'— ■>'-*"■+*»<'— ».)' ■ ■ • • ■ (535);

which equation having reference to the portion AE of the
beam, it is evident that when x=a,, -^-^tan, ,8.

:. — tan.,S=;Jfl(a-«,)'-i(ffl=-«,=)=i(«-«^(2«=-4<iffl,-<')

.... (536).

Substituting, therefore, for tan. S in equation (534), and
reducing, that equation becomes

f|=K+W«-.,)--i.- par).

, Google

i86 THE DUFLEXIClN OF A BEAM

Integratiug equation (535) "between the limits tt, and x, and
equation {531} between the limits 0 and x, and representing
the deflexion at C, and therefore tlie value of ^ at A, by D„

— (y— D,) = ■J^x'—}afyi—a,)'~\ia'— iff (a— «,)'{ tc— Ji-«,'+

ia\—iaa,{a—a^)'

~y=^\^'+ \^a{a-a:f~^a'\ic (538) ;

the former of which equations deterraines the neutral line
of the portion AE, and the latter that of the portion CA of
the beam. Substituting u, for x in the latter, and observing
that y then becomes D^ ; then Bubstitnting this value of D,
in the former equation, and reducing,

I>.=g|jil2<.(»-«,)--(«-».-)! .... (639);

— i/=^^.aj'— ^([(»!— (S,)'+Jffl j3(o;— a,y~ff°ja! .... (540).

The latter equation being that to the nentral line of the por-
tion AE of the beam, if -we substitute a in it for x, and
represent the ordinate of the neutral line at E by y,, ^Y&
shali obtain by reduction

^'■=1111 W-' + ^-X— «.)'-3»'! • • . ■ (Ml).

If ffl,=0, or if the loading commence at the point A of the
beam, the vahie of i/, will be found to be that already deter-
mined for the deflexion in this ease (equation 530).

!N"ow, representing the deflexion at E by D„ we have evi-
dently D,=D,— 2/,.

^,=-'~^{-&a' + tOaa, + a:\ (M2).

374. The cokdotons of the nEFLExios of a beam LoAnKc

BSIFOEMLT THROUGHOUT riS LENGTH, AND SUrPOETED AT

rra EXTRiMrriES A aud D, and at two point's B and C

BrrUATED AT EQUAL DISTANCES FKOM TEEM, AND IN THH
SAME JIOKIZONTAL STEAIGHT LINE.

Let AB=:«„ AD=:2a.

Let A be taken as the origin of the Go-ordinates ; let tlie

./Google

LOADED UNIFOEMLT.

].ii.3suie upon that point be
lepie^euted by P„ and the
pie bute upon E by Pj ; also
the load upon each unit of
the length of the beam by y..
If P be any point in the
neiiti il line to the portion AB
ot the beam, -whcme co-ordi-
nites ai'e x and y, the pres-
sures applied to AP, and in equibbrium, are the pressure
Pj at A, the load ftx supported by AP, and producing the
same effect as though it were collected over the centre of
that portion of die oeam, and the elastic forces developed
upon the transverse section of the beam at P ; whence it
foUowB (Art. 3600 by the principle of the equality of
moments, taiing P aa the point from which the moments
are measured, that

J this equation between the limits a„ and x, and
representing the inclination to the horizon of the tangent to
the neutral line at B by ^^

Integrating again between the limits 0 and x,

E%~ictaii.^,)=HiK''— ff»-iI'.(4»'-«» ■ ■ ■ (Si^).
"Whence observing that when ic=a„ 3^=0,

EI tan. /3,=ifJ'<~iPA'' (546).

Similarly observing, that if x and y be taken to represent
the co-ordmates of a point Q in the beam between B and C,
the pressures applied to AQ are the elastic forces upon the
at Q, the pressures P, and P, and the load ff-re, we

EI-,

=lM'-F,a^-P,(»'-

. (647).

Integrating this equation between the limits a, and x, and
observing that at the former limit the value of -rr is repre-
sented by tan. /S,, we have

./Google

4b8 the deflexion of a beam

^^ S-"°- ^-l =M»'-»,")-iP,(»'-».')-iP.(«-»,)'
.... C5i8).

Kow it ie evident that, since the props B and C are placed
eymmetrically, the lowest point of tlie beam, and therefore
of the neutral line, is in the middle, between B and C ; bo

that ;7 = Oi when a:=a. Making this substitution in equa-
tion (548),
-EI tan. fd,=i,^{a,'~a;)-i'P.(a.'-a,')~iV,{a-a,)' . . (54&).

Since, moreover, the resistances at 0 and D are equal to
those at B and A, and that the whole load upon tlie beam is
sustained by these four realBtaiices, we have

P, + P,=t*a (550).

Assuming a,=na, and eliminating P„ P„ tan. ^„ between
the equations (546), (549), and (550), we obtain

'^' 24EI" i 2n.—S

. (553).

24E[ I 2n-

Making a!=0 in equation (544); and observing that the cor-
responding value of -3^ is represented by tan. /3,, T,ve have

EI (tan. /3,-tan. /3,)= ~iva,'+^'P,a,,\

Substituting for tan. /3, and P^ their values from equations
(553) and (551), and reducing,

tan. ,e,= jg^ I g^^zrg \.... (o54).

Eepresentiog the greatest deflexions of the portions AB and

./Google

LOADED TJNIFOBMLT.

BC of the beiiTTL, respectively, by D; and D„ and by x^ the
distance from A at which the deflexion D, is attained, we
have, by equations {5ii) and (5i5),

-EItan.^,=^^{(.,--<)~JP,«-<) 1

EI(D,-£c,tan.,e,)=iKX-«>.)-iI'i(4^.'-«>.) f " "^ '"

The valne of D, is determined by eliminatii^ ;r, between
these equations, and eubstltuting the values of Pi and tan. ^,
from equations (551) and (553).

Integrating equation (5i8) between the lunits a, and a,
and observing that at the latter limit j/=D„ we have

EID,=:EI(a-ff,) tan. ^, + ifA{4(a'_0~a>-a,)} -

Subatitiiting in this equation for the values of tan. /3^, P^, P^,
and reducing, we obtain

D,=

■48EI(3-2rt)

j«.'—2n'— 871+6} (556)

Representing BO by %a^^ and observing that a^ = AE -
AB=(i— «a=(l— n)a,

fid,' «,'— 2ji'— 8«'+6
=l8EI ■ X3-3?i)Cl-tt.J' ■

. (557).

375. A -REAil, HAl'IXG .\ UNIFOHJI I,0.\D, SCl'l'Or.TED AT ^:\UU
EXTKEMITY, A>'D BY" A SINGLE STRUT IN THli MIDDLE.

If, in the preceding ai-tiele, a, be assumed equal to a, or
n=l, the two props B and
C will coincide m the centre ;
and the pressure P, upon
the single prop, resulting
from their coincidence, will
be represented by twice ihe
corresponding value of P, in
equation (552) ; we thus ob-
tain

P,= frj.((, P,— IfKt;

. (558).

./Google

490 THE DEFLEXIOH OF A BEAM

The distance x, of the point of gi'eatest deflexion of either
portion of the beam from its exti-emities A or D, and the
amoimt D, of that greatest deflexion, are determined from
equations (555). Making tan, /3,=0 in those equations,
substituting for P, its value, solving the former in respect to
iK„ and the latter in respect to D„ we obtain

3.^= ii-|^ffi=-431535« (559).

■^'^ 48EI - 48EI '•^''"J-

3'76. A BEAM "WHICH SUSTAINS A UHIFOEM LOAD THKOrGHOTTT
ITS WHOLE LENGTH, AND WHOSK KXTEEMlTmS AEE SO FHIMLT
IMBEDDED IN A SOLID StiSS OF MASONRY AS TO BECOME
EIGID,

Let the ratio of the lengths of the two portions AB and
AE of a beam, supported by two props (p. 487), be assumed
to be such as will satisfy the condition 5??.'— 16?i + 8=0; or,
solving this equation, let

n=5(4± V6) (561).

The value of tan. jS^ (equation 553) will tlien become
] zero; so that when this re-
1 lation obtains, the neutral
j line will, at the point B, be
\ parallel to the axis of the
\ abscissae ; or, in other words,
3 the tangent to the neutral
line at the point B will retain,
after the deilexion of the beam, the position which it had
before; i. e., its position will be that which it would have
retained if the beam had been, at that point, rigid. Now
this condition of rigidity is precisely that which results from
the ii^ertion of the beam at its exti-emities in a mass of
masonry, as shown in the accompanying figure; whence it
follows that the deflexion in the middle of the beam is the
same in the two cases. Taking, therefore, the negative sign
in equation (561), and substituting for n its value j(4— V6)
or '6202041 in equation (557), and observing tliat, in thai

./Google

: ANY XUMBEB OF 1

equation, 2a, represents the distance EC in tlie accompaay-
jng figure, we obtain

^ ^.=Hli «•

By a comparison of this equation with equation (530), it
appears that the dejlmwn of a learn austaming a presswi-
v/rdform^i disfy^imted over its whole length, mid having its
etcfff-mdHes prolonged and firml/y imbedded, is only one-fifth
of that whwh it would exhibit if its ecetren-dties were free.'''

If the masonry which rests upon each inch of the portion
AB of the beam be of the same weight aa that which rests
upon each inch of BO, the depth AB of the insertion of eacli
end should equal '63 of AE, or about three lou'hs of the
whole length of the beam.

377. Conditions of the equiUirium cf a, 'beam sw^orted at
any nmnherr of pomts artd deeded h/ g '

To simplify the investigation, let the points of support
ABO be supposed to be three
in number, and let the direc-
tions of the pressure bisect
j the distances between them ;
the same analysis which de-
termines the conditions of the
' equilibrium in this case will
be found applicable in the more general case. Let Pj, P^,
P„ be taken to represent the resistances of tlie several points
nf suppoi-t, a, and a, the distances betweeti them, P^ P, the
deflecting pressures, and x y the co-ordinates of any point in
the neutral line from the origin B. Substituting in equation

(500) for ^ itsvaUie-T^, and observing that in respect to the

portion BD of the beam 2Pp=Pj(^,— ai)— P,(ffi,— a), and
that in respect to the portion DA of the beam, spj>=
— l',(a,— k), we have for the differential equation to tlie
neutral line between B and D

* The following eiperimcEt wss made by Mr. Hatchar to verify this retult.
A strip of deal ^ in. bj-j% in. was supported with Its eitremities resting
loosely on roUere six feet iipart, and was observed to deflect 1-3 Inch in the
middle by its own weight. Tiie extremities were then made rigid b j confining
them between straight edges, and, the distance between the points of support
remaining the sanie, tlio deflesion was observed to be '22 ioeb. The theory
would, have given it "ii.

, Google

BEAM SUPPOETEn AT ANY NUMBER OF POINTS.

T^4i-=^-'-*''-''^-'^''^''~''> — f°'

between D and A

El|J = -P.(»-») . . . (564).

fiepreeenting by /3 the inclination of the tangent at B to the
axis of the abscisste, and integrating the fonner of these
eq^iiations twice between the limits 0 and x,

EI^=:^P,(»,a;-a^-P.(a,3j-i^)+EItatJ. /3 .... (565);

EIs/=iP,(i<«,a''-K)-iP,(ay-^')4-EIe tan. /3 . . . (566).

Substituting ^a^ for x in these equations, and representing
by D, the value of y, and by y the inclination to the horizon
of the tangent at the point I), we obtain

Eltan. r=;iP,<-|P,<+EItan.i3 .... (567),

EID,=:5-VP,<-iVPA'+iEI», tan. ,3 . . . . (368).
Integrating equation (564) between the limits -^ and x

EI^=-P,(a,a;— ia;')+EI tan. 7+|Pa°-

Eliminating tan. y between this eqnation and equation (667)
and reducing,

Elfc-P,(<x,a!-iic')+ETtan. /3+iPA' . . . (569).

Integrating again between the limits ~ and x, and elimi-
sating the value of D, from equation (568),
El2/=-iP,(«.a!'-430 + (EI tan. /3+|P,o,')at-JjP,a:' ■ (570)

Now it is evident thai the equation to the neutral line in
respect to the portion CE of the beam, will be detennined
by writing in the above equation P^ and P, for P, and P,
respectively.

Making this substitution in eqnation (570), and writing
—tan. ,S for +tau. ,8 in the resulting equation ; then assum-

,y Google

BEAM SDPPOKTED AT ANT NTJMBEK OF POINTS. 49o

ing ic=«, in equation (570), and K=ffl, in the equation tliiis
derived from it, and observing that y then becomes noro in
both, we obtain

0=— JP,«,'+ ' P,a,'+EIa, tan. ^,

0--iP.Q!,'+j^PA'— Elff, tan. ^.

Also, bj tlie general couditiona of the equilibrium of parallel
preasures (Art. 15.),

PA + iP.«,^PA + iP#„

P,+P, + P.=P,+P,.
Eliminating between these equations and the preceding, a»-
Burning o, + (Tj— ffl, and reducing, we obtain

_-r,a,(8«i + 5'^,)-3PA

-^'~ M\n.fi. .... V"'-';-

^P.a,(8«, + oaJ-3PA' ,5^2.

' Idaa, . . ■ . I, J.

By equation (5f!8),

Similarly,

By equation (567),

If ff, be substituted for ic in equation (569), and for P, and
tan. /3 their values from equationa (571) and (576) ; and if
the inclination of the tangent at A to the axis of a: be repre-
sented by /3^, we shall obtain by reduction

./Google

45i EEAM 8UPP0KTED i

'"•'^■=5SEEi Pa--Pa(2».+»,) } (378).

Similarly, if /3, represent the inclination of the tangent a(
C to the axis of ic,

'"•'"-saii i p.<-p.».(2».+«.) } (sw).

378. If tlie pressures P, and P., and also the distances «,
*nd a^, be equal,

P,=r.=,',.P,, P3=VP«tan.S— ^ ^- ^ -- '^ ^'*'

379. If the distances a,^ and o, be equal, and P,=3P„
P.=iP„ P.= VP„ P,=fP„ tan./3^_^' tan.^.^O*

380. If «,=a, and3P,=:lJiP„P,=0, P^:^VP"PB=i--P.-

* The foHowing esperlments were made by Mr. Halcher to verify this result.
The bar ACB, on. which the experiment was to be tried, wae supported on
knife edges of -wrought iron at A, C, and B, whose distances AO and CB were
e iOh five feet. The angles of the Itnife edges were 90°, and the edges were
oJed previous to the experiments. The weights were suspended at points D

and E intermediate between the points of support. In measuring the angles
of .lofleiion the instrument (which was a common weighted index-hand turn-
ing on a centre in front of a graduated arc) was placed so that the angle s
of vhe pavallelogram of wood carrying the arc was just over the Ifnife-edge B,
the side cd of the parallelogram resting on the deflected bar. This portion
gave the Single at the point of support.

1st Eiperiment. — A bar of wrought iron half an inch square, being loaded
at E with a weight of 18 lb. 18 oz., and at D with 52 11}. 3 oz., assumed a per-
fectly horizontal position at B, as Bbown by the needle. The proportion of
these weights is K'n r 1.

2d Experiment. — A bar '7 inches sq^uare, being loaded at E with a weight of
S';-3 lb., and at D with a weight of 112 lb.,, assumed a perfectly horizontal
position at B. The weights were in this experiment aoouralelj in the proper-

Sd Experiment. — A round bar, 'iD inch in diameter, being loaded at E with
SI'S lb-, and at D with 112 !b., showed a deviation from the horizontal position
at B araouutJGg to not more than 20'. The weights were in the proportion of

The influence of the ivoight of the bar is not talK'n into account.

, Google

L BEAJI DEFLKCTED BY

381. CUEVATURE OF A RECTAUGULAE BKAM, THE DIRECl'ION OP
THE DEFLECTING PKEBBUEE AND THE AMOUSIT OF THE DE-
FLEXION BEING ANT WHATKVEK.

Tbe moment of inertia I {Art. 868.) is to be taben, about
an asia perpendicular to the plane of deflexion, and passing
through the neutral line, the distance h of which neutral
line from th« centre of gravity of the section ie determined
by equation (499),

Now ^M representing (Art. 362.) the moment of inertia
of the rectangular section of the beam about an ax^ pass-
ing through its centre of gravity, it follows (Art. 79.) that
the moment I about an axis parallel to this passing through
a point at distance h from it is represented by

Substituting, therefore, the value of h from equation
(499),

I=^Ain.'a+-J55C....(580).

SubBtituting this value in equation (500), and redncing,
12F,Etiy,

B-13E"P,'iiin.'< + E'iV

. (581).

Draw ax parallel to the
position of the beam be-
fore deflexion; take this
line as the axis of the
abaci 8S£e and a as the
origin ; then^i =Em=E7i
, +nm.=MR cos, MEm+
aM. sin. Mflm.=y cos. Maw
-I- a; ein. Mirai.

Let, now, the inclination

DfflP, of the direction of

P, to the normal at a be

and the inclination TAat of the t;

represented by

the neutral line at a to era, by ^, ; then l£arn=^—{^,^-i^^.

.-.^,=2/ sin. (d,-(-S,)-HiBcos. (^ + /3,)-
Substituting this value of j), in the preceding eipiat.ion,

./Google

496 A BEAM DEFLECTED BY PEESiUKES.

1 12F.E^c jy eiT>. {6,+^,)+x cos. (a.+.S,)}

E~ 12ET,' sin. =fi+E'iV ^^^ ' '

■where 6 represents (Art. 355.) the inclination Jiqa of the
nonnal at the point E to the du'oction of P,.

382. Case m which the dejleadon of the l>emn, is small.

If the deflexion be small, and the inclination fl„ of tlie
direction of P, to the normal at its point of application, be

notgi-eater thanj; then y sin. (^+^,) is exceedingly small,

and may be neglected as compared with x cos. (^, + /3,}; in
this case, moreover, S is, for all positions of K, very nearly
ec[ual to ^,, Neglecting, therefore, ^3^ as exceedingly small,
we have

1 12F,Etocos.d,

E~12RT' sin. "fl.+E'JV

. . . (583).

Solving this equation, of two dimensions, in respect to =5-, ai

taking the greater root,

1 6P, ^__^^_____

^=£j^{iBcos. ^,+ ^si' COS. 'fi,— ^c'sin.'fij .... (584).

383. The "wokk expended tjpon the :

FOEM EECTAHOm^E BEAM, "WHEK THE IffiFLECTmG PKES-
SUEia ABE raCLIBED AT AWT ANGLE GKEATEK THAN HALF A
EIGHT ANGLE TO THE SURFACE OF THE EEAM.

If ii, represent work expended on the deflexion of the
portion AM of the beam, then (equation 505)

^'=2Ey r'^'

«' E p
out by equation (500) -y^pi ■ --p

./Google

:.v;^^l',f^dx (585).

^, 6P,

bj equation (584), observing that the deflexion being small.

p^=x COS. S, Tery nearly. Now the value of ^^ (equation

58i) becomes ivoposaible at the point wliere so cos. fl, becomes

less than — ^e sin. 6, : the curvature of the neutral line com-

menees therefore at that point, according to the hypotheses
on whieli that equation is founded. Assiuning, tiien, the

corresponding value —^o tan. ^, of ic to be represented by a;,,

the integral (equation 585) must be taken between the limits
», and Oi, instead of 0 and «, ;"

ZT'COS.i, /•; , . , -— i ,. ■ ,-y-. ST-, ,

:.u,= - ^, a -f p COS. 6,+x yx COS. 6,—f<r sm. d^\ax;:

:.^-i.= '-£l^' 'j<-^-— c'tan.'J. + (V-i^tan.'d^)^[n586).

And a similar expression being evidently obtained for the
work expended in the deflexion of the portion BM of the
beam, it foUowe, neglecting the term involving e° as exceed-
ingly small when compared with a', that the whole wort U,
expended upon the deflexion is represented by the equation

U,= 1^3 { P.' COS. \ 5a,' + «-ic= tan. M^*} +
P,'cos.''fl4ff/ + «— ^"tan. \y\ \

But if i, be taken to represent the inclination of Pj to the
normal to the surface of the beam, as fl, and fl, represent the
similar inclinations of P, and P„ then,, the deflexion being
small,

* Church's Int. Cal, Art. 14Sl
32

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TtEWIMimD BT PEESeUEES.

P,« COS. ^=Pa COS. S„ P,a COS. ^=P,(i, cos. ^,.

Eliminating P, and P, between these equations ar
preceding,

■p ' ^„. 'a 1 a

«,'K + k-ic'tan.X)^i I (587).

If tlie pressure P, be applied perpendiciilarly in the centre
of the beam, and the pressTires P, and P, he applied at its
extremities in directions equally inclined to its surface ; then
a,=a,=ia, S,=fl,=a, and S,=0, Substituting these valueH
in the preceding equations, and reducing,

384. The liseak deflexion of a

D, being taken as before (Art, 368.) to represent the de-
tEexion of the extremity A measured in a direction perpen-
.dicular to the smface of tlie beam, we have {Art. 62,)

M,=/P, COS. i,dJ),

dTt~dP, ' dl>;

But by equation (586), neglecting the term involv-ing c°,

^=jjj COB. ..!«.■ + (,. -Jo tan../)

Dividing Tiotii aides by P„ reducing, and integi-ating,

D,=|^.coa.(a< + (a,'-io" tan. ■»,)*! .... (589)
.Proceeding eimilarjy in respect to tlie detlection C perpen

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ANY ANGLE TO ITS SUKFACE. iv9

diciilar to the aiirfiice of the be^n at the point of application
of P„ we obtain from equation (5S7)

-p. 2P,C09J, ( ,(,,,, ,1. 1, \4, ,

^===^^^6^" i "' S''' +^'*- -*" ^^'^- ^'^^ +

a,'j<+{a/-~ic*tan. '(l,)*! I. . . . (590)

In the case in whicli F, and P, are eq^ually inclined to tlie
extremities of the beam and the direction of P, bisects it,
this etiuation becomes

;J8o. The wm'k exp&ndeii ■upon the iJt.flexion of n 'htam suh-
jected to the action of j>r>ss&ur(s a/ppUed to its eostremities,
and to a single intervemng poi/rvt, cmd also to the notion
of a system of parallel presmres imiformJy distributed
over its length.

Let «. represent the aggi-egate amount of the parallel

pressures distributed over eacli unit of the length of the
beam, and « their common inclination to the perpendicular
to the surface ; then'will f^ represent the aggregate of those
distributed uniformly over the surface DT, ana these will
manifestly produce the same effect as though they were
collected in the centre of DT, Their moment about the
point E is therefore represented by it-xi^ cos. o., or by ^m^
COS. a ; and the sum of the moments of the pressures applied
to AT is represented by (V^x cos. i^—ii>'X' cos. a). Substi-
tuting this value of the sum of the moments for P,^, in
equation (505), we obtain

1 n

'{FjX COS. ^, — it^" COB. a)'

, Google

L BEAM BY PEHfiSUKES.

8 he all perpeTidlcular to ths sv/rfacs of
the lea/m, ^,=0, a=0, and 1 is constant (equation 499);
whence we obtain, "by integration and redaction,

M.=^HP.'-iP,[^«,+iVi^V} (592).

If the pressure P, be applied in the centre of the beam,
Pi^iPj+ifia, and a,=^a, also the whole work If, of
deflecting the beam is equal to 2u^ ; whence, substituting

=mi^^^^'-^^^'^''+^'''"'^^-

. (598).

387. A EECTANGITLAK BEAM IS SITPOETED AT nS EXTEEMITIES
BY TWO PIXED 8TRFACE8, AND LOADED IN THE MIDDLE I PT
IB BEQUIEED TO DETEEMINE THE DEFLEXION, THE EKICTION
OF THE SUKFACES ON WHICH THE EXTBEMmffi BEST BEING
TAKEN INTO ACCOUNT.

It is evident that the work wliicli produces the deflexion

of the beam ia done upon it partly bv the deflecting pressure
P, and partly by the friction of the surface of the beam
apon the fixed points A and B, over which it moves whilst
in the act of deflecting. , Kepresenting by (p the limiting
tmgle of r^istance between the surface of the beam and
either of the surfaces npon wliich its extremity rests, the
friction Q, or Q, upon either extremity will be represented
by ^P tan. 9 ; and representing by s the length of the
curve oa or cb, and by 2a the horizontal distance between
the points of support ; the space through which the surface
of the beam woiiM have moved over each of its points of
support, if the point of support had been in the neutral line,
is represented by s—a, and therefore the whole work done
upon the beam by the iiiction of each point of support by

i tan. ff/Pife. Moreover, D representing the deflexion of

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THE SOLID OF THE STKONGEST FORM. 501

the boam under any pressure P, the whole work done liy P
is represented by iTdT). Substituting, therefore, for the
work expended upon the elastie forces opposed to the
deflexion of the beam ita value from equation (588), and ob-
serving that the directions of the resistances at A and B are
inclined to the normals at those points at angles equal to
the limiting angle of resistance, we have

/P^D + tan. .J Tds= -^^^,,eL' •

Put f YdD = f V^dV ; and/p,/s=/p|jc^P=:

-pa„ / V^d? by equation (521).

Substituting these values in the above equation, and dif-
ferentiating in respect to P, we have

■ptO) Pia' + ((t-'-|c'tan.V)l} Va"

Dividing by P, and integrating in respect to P,
Pi^°+K-^^tan/#| P^

§88. TilE SOLID OF THE STKONftEST EOEM WITH A GIVEN
QUANTITY OF MAT-EKIAL.

The strongest fonn which can be given to a solid body in
tlie formation of which a given quantity of material is to be
used, and to which the strain is to be applied under given
circumstances, is that form which renders tt equally liable to
TUfi/wre ai every point. So that when, by increasing the
strain to its utmost limit, the solid is brought into the state
bc^dering upon rupture at one point, it may be in the state
bordering upon rupture at every other point. For let it be
supposed to be constructed of any other form, so that its
rupture may be about to take place at one point when it is
not about to take place at another point, then may a portion
of the material evidently be removed from the second point
without placing the solid there in the state bordering upon

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502 'ITIE EUPTUEE Or A BAK,

ruptare, and added at the first point, so as to take it out
of tlie state bordering upon rupture at tiiat point ; and thus
the solid being no longer in tlie state Bordering upon
rupture at any point, may be made to bear a sti-ain gi'eater
than tbat which was before upon the point of brea£ng it,
and will have been rendered stronger than, it was before.
The first form was not therefore the strongest foi-ra of which
it could have been constructed with the given quantity of
material; nor is any form the strongest which does not
satisfy the condition of an equal Uabzhty to nature at every
point.

Tlie solid, constructed of tlio strongest form, with a given
quantity of a given material, so as to be of a given strength
under a given strain, is evidently that which can be con-
stt-ucted, of the same strength, with the least material ; so
that the strongest form is also the form of the greatest
economy of material.

ECFTCKB.

389. The rupture of a bar of wood or metal may take
place either by a strain or tension in the direction of its
length, to which is opposed its TENAcrrr ; or by a thivst or
compressing force in the direction of its length, to which is
opposed its resistance to Comfeession ; or each of these
forces of resistance may oppc«e themselves to its rupture
transversely, the one being called into operation on one side
of it, and the other on the other side, as in the case of

aTEAKSVEEBE StEAIN.

Tenacitt,

390. The tenacities of different materials ae they have
been determined by the best authorities, and by the mean
results of numerous experiments, will be found stated in a
table at the end of tliis volume. The unit of tenacity is that
opposed to the teai-ing asunder of a bar one s(iuare inch in
section, and is estimated in pounds. It is evident that the
tenacity of a fascile of n such bars placed side by side, or
of a single bar n square inches in section, would be equal
to n sucn units, or to n times the tenacity of one bar.

To find, therefore, the tenacity of a bar of any material
in povmdt^, multiply the number of square inches in its sec-

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ET7PTUKE OF A BA.K SUSPENDED VJiETICAlLT . oO£

by its tenacity per sc^uare iiiehj as shown by the

391. A BAR, COED, OE CnAIN 18 SUSPENDED VERTICALLY, CAE-
EYINU A "WEIGHT AT ITS EXTREMITY : TO DETEKMIXE THB
"CONDrriONB OF ITS EOPTUEB.

F'i/rst. Let the bar bo conceived to have a uniform section
represented in sq^uare inches by K ; let its length in inches
be L, the weight of each cubic inch f, the weight suspended
from its estremity "W", the tenacity of its material per square
ijich T,; and let it be supposed capable of bearing m, times
the strain to which it is subjected. The weight of the bar
will then be represented by /i-LK, and the strain upon ita
highest section by t»'L!K+ W'. lifow the strain on this section
is evidently greater than that on any other ; it is therefore at
this section that the rupture wiU t^e place. But the resist-
ance opposed to its rupture ia represented by Kt ; whence it
follows (since this resistance is m times the strain) that

KT=m(|xLK+W),

By which equation is determined the uniform section K of a
bar, cord, or chain, so that being of a given length it may be
capable of bearing a eti'ain m- times greater than that to
which it is actually subjected when suspended vertically.
The weight W, of the bar is represented by the formula

.■.W,=— ^- — V- (596 .

393. Secondly. Let the section of the rod be variable ; and
let this variation of the section be such that its strength, at
every ^omt, may be that which would cause it to bear,
witliout breaking, m times as great a strain as that which it
actually bears there. Let K represent this section at a point
whose distance from the extremity which carries the weight
W is a? ; then will the weight of the rod beneath that point

be represented by li>-'Kch ; or, supposing the spoeilic gravity

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5Ui RTIPTUEE OF A BAE STTSrENDliD VEElICALI.r,

of tlie matefial to bo every where the sarao, by ii-lKdio : also
the resistance of this section to mpture is Kt.

Differentiating tliis expression in respect to se, observing that
K is a function of ic, and dividing by Kr, we obtain

1 dK _ mih
K 5^ ~ T '

Integrating thia expression between the limits 0 and x, and
representing by Ko the area of the lowest section of the rod,

, K. mil. -.^ -rr *

log. ■^=-^x; /, K=Koe t

But the strain sustained by the section Ko is "W", therefore
KoT=mW ;

. (597).

The whole weight W^ of the rod, cord, or chain, is repre-
sented by the formnla

■W,=/&fe=!^^.T»'&,=w(.— -l) . . . (598).

A rope or chain, constructed according to these conditions,
is evidently as strong as the rope or chain of uniform section
whose weight W, is determined by equation (596), the vahxe
of m being taken the same in both cases. The saving of ma-
terial effected by giving to the cord or chain a section vary-
ing according to the law determined by equation (598) ia
represented by W,— W„ or by the formula

T— mff-L

-■\VL"V^-1 (599).

" Church's Int. Cal.

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THE BUSrENSION I

ibrimn of a loaded

ThK aUSPEKSIClS ]

393. General conditions of the •
chavn.

Let AEH represent a chain or cord hanging freely from
two fixed points A and H,
and having certain weights
■w,, Ws, w„ &c., Biispended by
rods or cords from giren
points B, 0, D, &c., m its
length, Tliroiigh the lowest
point E of the chain draw
tlie vertical Ea, containing
as many equal parts as tliere
are unite m the weight of
the chain between E and any
point of suspension B, to-
gether with the suspending
rods attached to it, and the weights which they severally
cany ; draw aP parallel to the direction of a tangent to the
cnrve at B, and produce the tangent at E to meet aP in P ;
then will aF and EP contain as many eqnal parts as there
are units in the tensions at B and E respectively ; and if EJ
and Ee be taken to represent the whole weights sustained by
EC and ED, and P5 and Po he joined, these lines will in
like manner represent the tensions upon the points C and X>.
Por the pressures applied to EB, and in equilibiinm, heing
the weight of the chain, the weights of the suspending rods,
the weights attached to the roifi, and the tensions upon B
and E, the pi-inciple of tlie polygon of pressures (Art. 9.)
obtains in respect to these pressures. Now the Hues drawn
to complete this polygon, parallel to the wdghta, ioi-ro.
together the vertical line E«, and the polygon (resolving
iteelf into a triangle) is completed hy the lines dP and EP
drawn parallel to the tendons upon E and E. Each line
contains, tlierefore, as many eqnal parts (A. !.'. 9.) as tiiere
are units in the corresponding tension. Also, the pressures
applied to the portion EO ot the curve, heing the weights
whose aggregate is represented hy E5, and tlie tensions upon
E and O, of which the former is represented in direction
and amonnt by EP, it follows (Art. 9.) that the latter is
represented also in direction and amount by the line P&,

./Google

506

which completes the triangle aBh\ so tliat JP is parallel to
the tangent at C.

Ill like manner it is evident that tlie tension upon D is
represente.d in magnitude and direction by cP ; so that eP is
parallel to the tangent to the cJirve at D.

The

394. ^ a, eham of uniform section ie suspended freely
lehoeen two fixed points A and B, leing acted upon ly no
other pressures tMun the weights of its parts, then it wiU
assume the geometrical form of a ourve called the

Let PT be a tangent to any point P of the curve inter-
secting the vertical CD passing through its lowest point D

in T ; draw the horizontal line DM intersecting PT in Q ;
take this line as the axis of the abscissa ; and let DM =x,
MP=y, DP=fi, weight of each unit in tiie length of the
chain =:**, tension at D=c. Now DT being taken to repre-
sent the weight i^s of DP, it has been shown (Art. 393.)
that DQ win represent the tension o at D, and TQ that
at P.

= ti;n. PQM = tan. DQT=^

DT

dy ^s

. (600).

Integi'ating be-

,y Google

rHE CATIiNAEY. 507

tween the timits 0 and s,* and observing that when 5=0,
By addition and reduction,

-7 ~r\ («02).

Substituting this value for s in equation (600), and inte-
gra,ting between the limits 0 and s:,

3/=i-( T ~r l=i-( 25 -^] (603);

which ig the equation to the catenary.

{0) on the lowest ^owii of the

Let 28 represent the whole length of the chain, and 2a
the horizontal distance between the points of attachment.
Now when !>i=a, 8=S ; therefore (equation 602),

■ (604) ;

for which expression the value of c may be determined by
approximation.

396. The tension at amy ^oint of the chain.

The tension T at P \& represented by TQ= yW+M";

* Chuvch'e Int. Cal. Art. 114.

./Google

Now the value of c has been determined in the preceding
article ; the tension upon any point of the chain whose dis-
tance from its lowest point is s is therefore known.

397. The iaicUnation of the eu/roe to tlie vertical at any
point.

Let ( represent tills inclination, then cot. '—-^ ;

.-.{equation 600) cot. i-=i\ c e i {^^^)-

The inclination may be determined without having first

determined the value of c, by substituting cot. ( for — in

equation (601) ; we thus obtain, writing also a and S for x
and s,

^=tan. ( log. (cot. t + cosec, t)=tan, ( log, cot. -^t;

.-. —tan. t log. ^ tan. ^1=5 (607).

This equation may readily be solved by approximation ; and
the value of e may then be determined by the equation
t;=^iS tan. I.

198. A ehmn of gwen length heing suspended between two
gwen points in the same horizontal line : to determine the
d&pth of the knoest point beneath the points of attach/iiiieni ;
and, oonverseh/, to determine the length of the ohmn whose
lowest point shaU hang at a given depth lelow its j>oints
of attaehm&ni.

The eame notation being taken as before,
ds

= (i + S)"*=(in^)" = («-+.^v)-v..

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THE "CATENAKT. 509

Integrating between tlie limits 0 and s, and observing that
y=:0 when 5=0,

S=l\{<^+r-V)'-4 (608).

Solving tliis eq^uation in respect to 8,

'=yy(y+j] («™)-

If H represent the depth of the lowest point, or the versed
sine of the curve, then y=Ti when s==S.

K=-\{o-' + i^'Sy-o} (610).

S=:j/h(h+~) (6U).

399. The cmAre of gravity of the oaiena/tij.

If G represent the height of the centre of gravity above
the lowest point, we have (Art. 33.)

S.G=/y&=/j^*r.

Subatiimting, therefore, for y and -=- their values from equa
tions (602) and (603), we have

S.

V /. +. +2-2*. +. 'i

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THE SUSPEKSION BEIDOE

T3ut by equation (604) S^ipfT ~i^\ and by equation

H-- =

.•.SG=i S (H-

-i)\-

. (613).

400 The subPEHsioN beidge op skeatest ;

WEIGHT OF 1"HB SCSPEHDHsG BODS BETNG NEOIKIED

Let ADB ie]>re?ejit the chim, TF the loiJ ■waj , and let

the weight ot i bai ot the matenal nt the chiin one squire
inch in section and one foot long, be represented by ii-,, the
weight of each foot in the length of the road-way by (>■„ the
aggregate section of the chains at any point P (in sqnare
inches) by K, the co-ordinates DM and MP of P by ar and y,
and the length of the portion DP of the chain by s. Then.

will the weight of DP be represented by f)-, / Kds, and the
weight of the portion CM of the roadway by f*^ ; so that
the whole load (u) borne by the portion DP of the chain
will be represented (neglecting the weight of the suspending
rods) by

!>■, I Krfs-f iijic, ;, w=Hi / .

Kds+

. (61i).

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OF GKEAT1",ST BTKK^'(5TII. 511

Let this load (m), supported by the portion DP of the
chain, be represented by the line Da, and draw Dp in the
direetioa of a tangent at D, representing on the same scale
the tension o at that point ; then will ap be parallel to a
tangent to the chain at P (Art. 393).

" dx 0

Now let it be assumed that the aggregate section of the
chains is made so to vary its dimensions, that their strength
may at every point be equal to m times the strain which
they have there to sustain. But this strain is represented in
magnitude by the line ap (Art. 893.), or by (c'+w')*} if,
therefore, r he taken to represent the tenacity of the mate-
rial of the chain, per square inch of the section, then

KT=m{e'+uy (616).

Therefore 'K.r=mcll + ^J = mc (l + -^A (equation 616)

=mo^- : therefore ^-=-- . Also / Kd^= f K-r(fe=
iUe' (Ix mo J J aas

■ — / K'(&^=' — / {e' + u')dx (equation 616);

.•.(equation 614)w= — -I {c'+vi')dx+!i.,x.

Differentiating in respect to x, and observing that -5- =

du dy vidu , . „__

-j- -^=~ T- (equation 615), we have

ay ase G dy

du u du mfi, , , , m!^, l „ , titij,,*

dx c dy Te^ ' ' re \ mi^J '

_ re /■ du _JL. r '"'^'^

""ntfi-J rcii-, ~ rm'-.J „ , t<7u.„'

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512 THE SU3PEHSI0N BEID&B

Integrating these expressions,* we obtain

^^i!LU+:£h\-\^„,~'h+-^r\, (61,).

V-=-. WS- 1 ~ — ■ r

I e'+ — - \

\ m,D. J

Substituting in this equation the value of m given by the
preceding equation, and reducing,

wMoh is the equation to the suspension chain of uniform
strength, and therefore of the gkeatest strength with a

GIVEN QUANriTY OF MATERIAL.

401. To determine the vaHation if the section K of the
chain of the suspension bridge of the greatest strength.

Let the value of u determiiJied by equation (617) he sub-
stituted in equation (616) ; we shall tlius obtain by reduction

It is evident from this expression that the area of the sec-
tion of the chains, of the suspension hiidge of uniform
strength, and therefore of the greatest economy of material,
increases from the lowest point towards the points of suspen
Bion, where it is greatest.

• Church's Int. Cal. Art. ISS, Case IV,

f —= — ; ,■.!=— /'Kdx. Now the function K (equation aielmajhe

integrated in veepeol; to xhy known rules of the integral ealculasi the Taluo
of s may thecefoca be deMrmined in terms of x, niid thenofi tlio length ill
terms of the span. The formula is omitted bj reason of its length.
Church's Int. Cal. Art. 129, Case II.

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OF GKKATEST SlIlKKGTi

402. To detei'miTie the weight "W of tJie c/imn of ti
don bridge of the greaiest strength.

Let it be observed tJiat W=(ii / Kds=u--ii.^x (equation

014) ; substituting tbe value of u from equation (61T), we
have

W=«(l+iIi-\'tan.j=^(n-"i)'^l-^»,. . . (620).

403. To determine tJie tension o upon the lowest ^oint D of
the chain of tmiform strength.

Let H be taken to represent the deptli of tbe lowest point
D, beneath tlie points of suspension, and 2a the liorizontal
distance of those points : and let it be observed that H and
a are corresponding values of y and a; (equation 618) ;

.■,11=^ — log. sec. < ■ —
Solving this equation in respect to <?

1+:^

. (621).

404. The suspiqtsion BEmoK of geeatest steehgth, thb

WEIGHT OF THE SUSPENDING EODS BEtNG TAKEN INTO AO-
COUNT.

Conceive the suspending rods to be replaced by a con-

,y Google

514 THE SUSPENSION BKIDQE

tinuons flexible lamina or plate connecting tlie roadway with
the chain, and of ench a uniform thickness that the material
contained in it may be precisely equal in weight to the mar
terial of the suspending rods. It is evident that the condi-
tions of the equilibrium will, on this hypothesis, he very
nearly the same as in the actual case. Let n, i-epresent the

weight of each square foot of this plate, then will (*, / ydss

represent the weight of that portion of it which is suspended
from the portion DP of the chain, and the whole load « upon
that portion of the chain will be represented by

'M=^t,/li;o's+H'iiK+c.,/s'(& .... (622).

It may be shown, as before {Art, 400.), that

|4K.=,»(.+«-)' (0.3).

/E:«?s=—_/(<r' + «')<:&. Substituting in equation (622),

du udu Till*,., , ,. , , //.-i.N

Transposing, roducing, and a

^'=:a (625);

A linear equation in m', the iutegi-ation of wliich, by a well
Inown method gives

■■'iof\',y-\-aC-\-v^e dy-\-Q.^

Assuming the length of the shortest connecting rod BO to
be represented by h, integrating between the limits l and y,
and observing that when y=h, w=0.

* Church's Int. Cal Art, 176.

./Google

' STRENGTir. 5jO

:.^'=-\ is-ll^ -y)+(2;+«''+f'=)(^ -i)}-(62(i).

Subetitnting this value of v? in eqnatioii (£!23), and
reducing,

K=^-^ I i^+i^J> + o.<;+!s.,js ~^2/-2^-i^i [ -(627);

by which expression the variation of the section of the chain
of uniform strength ia determined.

Differentiating the equation 'f'=-~ in respect to x, and
enbstituting for -r- its value from equation (634).

iSnbstitutiTig for u' its value from equation (626),

Multiplying both sides of this equation by -^^ and integrat-
ing between the limits h and y, observing that when y=h,

ax '

"(l)"=(£+^-*+"+^-)('"°""'-i)-''-(!'-»)-

Now let it be observed, tliat the value of r, being in all
practical cases exceedingly great as compared with the
values of f^j and m, the value of a (equation 625) ia exceed-
ingly araali ; so that we may, without sensible error, assume
thc«6 terms of the series ^"'s-'') which involve powers of
2a(y— 6) above the first, to vanish as compared with unity

* Ckurth'e Ii\t. ChL Art. 140.

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THE 8USPENSI0X TlilDGE.

This supposition "being made, we have e^"**"*)— l=2a(j'— 5),
whence, by substitution and reduction,

'(2)'=^(^-'+"+''-> (y-*'-

Extracting tbe square root of botli sides, trausposing, and
integrating.

\|Aj&+aO+(A,

(^-i)-

the equation to a parabola whose vertex is in D, and its
axis vertical.*

The values a and H of ic and y at the points of siispension
being substituted in this equation, and it being solved in
1 espeet to c, we obtain

«=(ra5Sis)»' «^

liy which expression the tension c upon the lowest point of
the curve ia determined, and thence the length y of the sus-
pending rod at any given distance a) from die centre of the
span, by equation (628), and the section K of the chain at
that point by equation (637), which last equation gives by a
I'eduction similar to the above

405. The section of the chains ieina of 'wniform dimensions,
as in the common suspermon bndge^ it is requi/red to
determme the eonditi.ons of the eguilihHum.^

The weight of the suspending rods being neglected, and
the same notation being adopted as in tiie preceding arti-
cles, except that jJ', is taken to represent the weight of one
foot in the length of the chains instead of a bar one square
inch in section, we have by equation (614), since K is here
constant,

u^^^s + v^x . (631).

" Church's Analyt, Geom. Art. 191.

f This problem appears first to have been iiiTcstigai«d by Mr. Hodgkinson
in the fifth volnme of the Manchester Transactions; his investigation extends
\i> the cose in which the influeneo of the weights of tbe suspending rods \a
jjicluded.

, Google

THE GOirMON SUSI'ENSION BGIDGIE. oil

Differentiating tliis eqiiation in respect to x, and oliscrving
that £= (l + JJ)^= (l + -^if (eauatioa 615), aiid that

du_du <it/_dii u _ I ^'\^ ,
d^~ diy dxT' fly e"~ '\ cv "

/cdu _ /* ^t^i^

Tlie foiiner of these ec[nation3 may be rationahsed by
asBuming (c' + w' )^ = c + 2«, and the latter by assuming
(o'+m')*=2 ; there will thus be obtained by reduction

"-'Y (l-«')K^+^)+(^-l».>■|'"~/ i».2+w'

The latter equation may be placed under the torm

■which expression 'being integrated and its value substituted
for 3, we obtain

j=ll(..+„y_„_i?il„g. !^K±^)^»l...(632).

The method of rational fractions (Church's IvAeg. Cole.
Art. 135) being applied to the functiou under the integral
sign in the former equation, it becomes

The integi-al in the first tenn in this expression is repre.
sented by \ log.g ( t^^I , and that of tlie second term by

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EUPTUEE BY COMPKKSSION.

according as f^, is gi'eater or lees than (*,, or according as the
weiglit of each foot in the length of the chauis is greater or
less tlian the weiglit of each foot in the length of the road-
way-

Substituting for s its value, we obtain, therefore, in the
two caBes,

_2fh_

log-

-(^;)*]('+5)*-al

o».+p.)i'H-(^i-/'t)M(^'+°')^-4 1

1£ the given values, a and H, of x and y at the points of
suspension, be substituted in equations (633) and (632),
equations wiU be obtained, whence the value of the constant
c and of « at the points of suspension may be determined by
approximation. A series of values of u, diminishing from
the value thus found to zero, being substituted in equations
(633) and (632), as many corresponding values of x and y
will then become known. The curve ot the chains may thus
be laid down with any required degree of accuracy.

This common method of constimetiou, which assigns a
uniform section to the chains, is evidently false in principle;
the strength of a bridge, the section of whose chains varied
according to tlie law established in Art. 401. (equation 619),
would be tar greater, the same quantity of iron being
employed in its constniction.

EUPTTJKE BY COUPKESSION.

406. It results from the experiments of Mr. Eaton Hodg-
kinson,* on tlie compression of short columns of differant
heights but of equal sections, fii'St, that after a certain height
is passed tlie crushing pressure remains the same, as the

• Seventb Report of the Bi-iiish AssociHtion of Science,

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SDPTHEE BT

heights are increased, until another lieight is attained, when
tliey begin to breai ; not as they have done before, by the
sliding of one portion npon a subjacent portion, bnt by
Ijending. Secondly, tliat the plane of rupture is always
inclined at the same constant angle to the base of the
column, when its height is between these limits. These two
facts explain one another ; for if K represent tlie transverse
section of the column in square inches, and a the constant
inclination of tlie plane of rupture to the hase, then will
K. sec. a represent tne area of the plane of rupture. So that
if 7 represent the resistance opposed, by the coherence of
the material, to the slidine of one square inch upon the sur-
face of another,* then wilT/K sec. a represent the resistance
which is ovei-come in the rupture of tlie column, so long as
its height lies between the supposed limits ; which resist-
ance being constant, the pressure applied upon the summit
of the column to overcome it must evidently be constant.
m Let this pressure be represented by P, and let CD
* be the plane of rupture. Now it is evident that
the inchnation of the direction of P to the perpen-
dicular QR to the surface of the plane, or its
0 equal, the inclination a of CD to tlie base of the
'' column, must be greater than the limiting angle
of resistance of the surfaces ; if it were not, then
woiild no pressure applied in the direction of P
De sufficient to cause the one surtaoe to slide npon the other,
even if a separation of tlie sui-faces were produced along
that plane.

Let P be resolved into two other pressures, whose direc-
tions ai-e perpendicular and parallel to the plane of i-upture ;
the former will be represented by P cos. a, and the friction
resulting from it by P cos. a tan. <p ; and the latter, repre-
sented by P sin. a, will, when rupture is about to take place,
be precisely equal to the coherence K7 sec. a. of the plane of
rupture increased by its friction P cos. a tan. o, or P sin.
K=K7 see. a + P cos. a tan, p, whence by reduction

p^ KrcoB.-p _ ^Kycos. <p ,^^^.

sin. (a— <p) COS. a sin, (3a— 9) — sin. ip ■*■■'• r

It is evident from this expression that if the coherence of
the material were the same in all directions, or if the unit of

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520 THE PLANE OF EUrTDEE.

coherence y opposed to the slidiug of one portion of the
mass upon another were accurately the same in every direc-
tion in which the plane CD may be imagined to intersect
the mass, then would the plane of actual mptnre be inclined
to the base at an angle represented by tlie formula

. (635) ;

since the value of P would in this case be (ecLuation 634)
a minimum when sin. (3a— ^) is a maximum, or when

2a— ip=-, or a.=--\-^ ; whence it follows that a plane in-
clined to the base at that angle is that plane along which the
ruptm-e will firet take place, as P is ^-adually inci'eased be-
yond the limits of resistanee.

The actual inclination of the plane of rupture was found
in the experiments of Mr. Hodgtinson to vary with the ma-
terial of the column. In cast iron, for instance, it varied
according to the quality of the iron from 48° to 58°*, and
was difterent in different species. By this dependence of
the angle of rupture upon the nature of the material, it is
proved that the value of the modulus of sliding coherence
y is not the same for every direction of the plane of rup-
ture, or that the value of 9 varies gi'eatly in different quali-
ties of cast iron.

Solving equation (634) in respect to 7 we obtain

p

y=w sin. (m— 9) COS. a see, 9 (636) ;

fi'om which expression tlie value of the modulus 7 may be
determined in respect to any material whose limiting angle
of resistance <p is known, the force P producing mptiire,
imder the circumstances supposed, being observed, and alsc
tlie angle of rapture.^

The section op kuptuee tn a beam.
407. "When a beam is deilected under a transverse strain,

■ Serenth Keport of British Asisoclation, p. B49.

\ A detMled stalemeiil of the reanlta obtained in the experiments of Mr.
Hodgkinson on tbia subject is contained in tlie Appendix to the " Illuatradona
of MeoUamcB" by the authoi' of this work.

, Google

GKNEEAL CONDITIOIfS OF EDlrTDRB. o21

the inateTJal on that side of it on whicli it sustains the strain
is compressed, and the matenal on the opposite side
extended. That imaginary surface which separates the
compressed from the extended portion of the mateiial is
called its neutral surface (Art. 354.), and its position has
been determined under all the ordinary cii-cnmstances of
■flexure. That which constitntes the strength of a beam is
the resistance of its material to compreBsioii on the one side
of its neutral surface, and to extension on the other ; so that
if eitha' of these yield the beam will be broken.

The section of Twpture is that transverse section of the
beam about wliich, m its state bordering upon rupture, it is
the most extended, if it be about to yield by the extension
of ite material, or the most compressed if about to yield by
the compression of ite material.

In a plasmatic beam, or a beam of uniform dimensions, it
is evidentiy that section which passes through the point of
greatest cnrvature of the neutral line, or the point in
respect to which the radius of curyatnre of the neutral line
is tiie least, or its reciprocal tlie greatest.

General conditions oy the

408. Let PQ be the section of rupture in a beam sustain-
ing any given pressures, whose
resultants are represented, if
they be more in number than
three, by the thi'ee pressures P„
Pa, P,. Let tlie beam be upon
the point of breaking by the
yielding of its material to exten-
sion at the point of greatest ex-
V is tension P ; and let E represent,

in the state of the beam border-
ing upon rupture, the iutereection of the neutral surface
with the section of rupture ; which intoi'seetion being in
the case of rectangular beams a straight line, and being in
fact the neutral axis, in that particular position which is
assumed by it when the beam is brought into it« state bor
dering upon rupture, may be called me axis of rwp^re ^
aK the area in square inches of any element of the section
of rupture, whose perpendiculai' distance ti-om tlie axis of
rupture H is represented by p; 8 the resistance in pounds

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523 GENEEAX CONDmONS OF HCPTUKE

opi>osi:d to the nipture of each square inch of the section at
r ; e, and e, the distaneea PE and QR in inches.

The forces opposed per square inch to the extension and
compreseion of the material at different points of the sec-
tion of rupture are to one another as their several pei-pen-
dicular distances from the axis of rapture, if the elasticity
of the material be supposed to remain perfect throughout
the section of rupture, up to the period of rapture.

Now at the distance e, the force thus opposed to the
extension of the material is represented per square inch hy
S ; at the distance p the elastic force opposed to the exten-
sion or compression of the material (according as that
distance is measured on the extended or coinpi'essed side), ia

therefore represented per square inch by —p, and the elastic

force tlius developed upon the clement ^K of tl^e section o(

rupture by — p^^K, so that the moment of this elastic force

about R is represented by — p'^K, and the sum of the mo-
ments of all the elastic forces upon the section of rupture
about the axis of rupture by — Sp'AK ;* or representing the

moment of inertia of the section of rapture about the axis
of rupture by I, the sum of the moments of the elastic
forces upon the section of rupture about its axis of rupture

is represented, at the instant of rupture, by — -f Now the

elastic forces developed upon PQ are in equilibrium with
the pressures applied to either of the poiiiionB APQD or
EPQO, into which the beam is divided by that section ; the
sum of theh' moments about tiie point P is therefore equal
to the moment of R, about that point. Representing,
therefore, byjp, the perpendicular" let fall from the point R
upon the direction of P,, we have

* It will be observed, oa in Art. SBS., that the elastic forces of e;
Hid those of compreBaon tend to turn the surface of rupture in the samo
lireetion about the axis of rupture.

I This expression is called by the Frenoli wiiters tJie moment of rupture j
(lie beam is of greater or leas strength under given ciruumstajices accorrting
as it has a greater or less value.

, Google

1'.?.=^ (637)-

409, If the deflexion be small in the state bordering upon
mpture, and the directions of all the deflecting pressures be
perpendicular to the anrfaoe of the beam, the axis of rupture
passes through the centre of gravity of the section, and the
value of Cj is known. Where these conditions do not obtain,
the value of o, might be determined by the principles laid
down in Arts. 355. and 381. This determmation would,
however, leave the tlicory of the rupture of beams still in-
complete in one important particular. The elasticity of the
material has been supposed to remain perfect, at every point
of the section of rupture, up to the instant when rupture is
about to take place. Now it is to be observed, that by rea-
son of its greater extension about the point P than at any
other point of the section of I'upture, the elastic limits ai'e
there passed before ruptm-e takes place, and before they are
attained at points neai'er to tlie axis of rupture ; the forces
opposed to the extension of the material cannot therefore be
assumed to vaiy, at aU points of PE, accurately as their dis-'
tances from the point R, in that state of the eq^uilibrium of
tlie beam which immediately precedes its rapture ; and the
sum of their moments cannot tlierefore be assumed to be ac-
curately represented by the expression —-. Tliis remai-k af-
fects, moreover, the determination of the values of A and K
(Arts. 355. and 381.), and therefore the value of c.

To determine the iniluence upon the conditions of ruptui-e
hy transverse strain of that unknown direction of the insistent
pressures, and that variation from the law of perfect elasti-
raty which belongs to the state bordering upon rupture, we
must fall back upon experiment. From this it has resulted,
m respect to Teota/ngvlar beaTM, that the error produced hy
these different causes in etiuation (637) will be corrected if
a value be assigned to c, bearing, for each given material, a
constant ratio to the distance of the point Pfrom tlje centre
of gravity of the section of mpture ; so that a representing
the depth of a rectangular beam, the error will be corrected,
in respect to a beam of any matei-ial, by assigning to c, the
value tn^c, where tjj- is a certain constant dependent upon
the nature of the matei'ial. It is evident that this cor-
rection is equivalent to assuming c,=^, and assigning
\a S tlie vahie -S instead of that which it has hitherto

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524

". coxrrnoNS of EurTUEr

been supposed to represent, viz. the tejiacity per equare inch
of the material of the beam.

It is cnstomaiy to make this assumption. The values of S
corresponding to it have been determined, by experiment,
in respect to the materials chiefly used in construction, and
wiU be found iti a table at the end of this work. It is fo
theae tables that the values represented by S in all subse-
quent formulse are to be referred.

410. From the remai"ks contained in the preceding article,
it is not difficult to conceive the existence of some ifirect re-
lation between the conditions of rapture by transverse and by
longitudinal strain. Such a relation of tlie simplest kind ap-
pears recently to have been discovered by the experiments
of Mr. E. Hodgkinson*, extending to tlie conditions of rup-
ture by compression, and common to all the different varie-
ties of material included under each of the following great
divisions — timber, cast iron, stone, glass.

The following tables contain the summary given by Mr.
Hodgkinson of his results ;-—

rieapi-iption ot MjiterluJ.

i;sX£Zz

Mean Tensile
StitngUi per Square

Mean TcmByerse
attength of ■ Bsr
1 Inoli Sqnare and

Timber ....

Caat-iroQ

Stone, including marble -

aiaea (plate and crown) -

1000
1000
1000
1000

leoo

163
100
123

85-1
19'8
9'S
!0'

The following table shows the uniformity of this ratio in
respect to dift'erent varieties of the same material : —

Mean leo-ile

I 1^1 Sqiire^M

Slreugtli pBr Square

Strength per SquBrt

IniLi.

111 ell.

Black marble -

1000

143

10-1

1000

84

I0'6

Roehaaleflngstoiie-

lOOO

104

9-9

High Moorstone -

lOOO

100

Yoikihire flag

1000

96

Stoae fiom Little Hulton,
near Bolton

I 1000

70

8-8 J

, Google

i 9TK0NGICST FORM (

411. The bteongest form of section at any given point

IN THK I.ILNGTn OF THE BE All.

Since the extension and the compression of the n

are the greatest at those points whick are most distant from
the neutral axes of the section, it is evident tliat the mate-
rial cannot he in the state bordering upon mpture at every
point of the section at the same instant (Art. 388.), unless all
the material of tlie compressed side be collected at tlie smne
Mst(mc6 from the neutral axis, and likewise all the material
of the extended aide, or nnless the material of the extended
side and the material of the compressed side be respectively
collected into two geometrical lines parallel to the neutral
axis : a distribution manifestly impossible, since it would
produce an entire separation of tlie two sides of the beam.

The nearest practicable approach to this form of section is
tliat represented in the accompanying figure, where the
mateiial is shown collected in two thm but wide flanges,
united by a narrow rib.

I — . . — J That which constitutes the strength of the
beam being the resistance of its material to com-
pression on the one side of its neutral axis, and
its resistance to extension on the other side, it is
evidently (Art. 388.) a second condition of the

strongest form of any given section that when

the beam is about to break across that section by
extension on the one side, it may be about to break by com-
pression on the other. So long, therefore, as the distribution
of tlie material is not such as that the compressed and
extended sides would yield together, the strongest foEm of
section is not attained. Hence it is apparent that the
strongest form of the section collects the greater quantity
of tlie material on the compressed or the extetxled side of
the beam, according as the resistance of the material to
compression or to extension is the lefts, Wliere the material
of the beam is cast iron*, whose resistance to extension is
greatly less than its resistance to compression, it is evident
fliat tlie greater portion of the material must be collected on
the extended side.
Tims, then, it follows, from the preceding condition and

* It IS onlj in. oast iron beams that it U custflmary to seek an economy of
the material In the strength of the section of the beam ; the aamo priueiple of
ocoaomy is euvely, however, applicable to beams of wood.

, Google

THE STKONGEaT FORM OF

this, thai; tlie strongest form of section in a east iron beam ie
that by which the material is collected into two uneqtial
flanges joined by a rib, the greater flange being on the
extended side ; and the proportion of this inequality of the
flanges being just such as to make up for the inequality of
the resistances of the material to rupture by extension and
compression respectively.

Mr. Hodgkinsoii, to ■whom tliis suggestion is dne, has
directed a series of experiments to the determination of that
proportion of the flanges by which tlie strongest foi-m of
section is obtained.*

The details of these experiments are found in the following
table :—

Ex

"ci^™t.

••s.;s.ss,"'

SectTodln^quare

£

ngthpcrS[|uare

1

1 to T

2-82

2B6S

3

l-S'J

2667

S'()2

2737

6

I to 4-5
1 to 5'S

3-37

5'0

8848

6

1 to 6'1

6-4

4075

In the firet flye experiments each beam broke by tiie tear-
ing asunder of the lower flange. The distribution by which
both were about to yield together — that is, the strongest
distribution — was not therefore up to that period reached.
At length, however, in the last experiment, the beam yielded
by the compression of the upper flange. In this experiment,
therefore, the ujyper flange was the weakest ; in the one be-
fore it, the lower flange was the weakest. For a form
between the two, therefore, the flanges were of equal strength
to resist extension and compression respectively ; and this
was the strongest form of section (Art. 388.).

In this strongest form the lower flange had six times the
material of tlio upper. It ia represented in the accompany-
ing figure.

In the best fona of cast iron beam or
gii'der nsed before these experiments,
there was never attained a strength of
more than 2885 lbs. per square inch of
section. There was, therefore, by this
form, a gain of 1190 lbs. per square inch
zn of the section, or of fths the strength of
the beam.

./Google

412, The siicnoN of euptuee.

The conditions of rupture temg determined in respect to
mvy section of the beam by equation (637), it is evident that
the particular section across wliich rupture will actually take

Elace is that in respect to wliich equation (637) is first satis-
ed, as P, is continually increased ; or that section in respect
to whicli the formula

(638)

Pfii
is the least.

K the beam be loaded along its whole length, and x repre-
sent the distance of any section from the extremity at which
the load commences, and |i the load on each foot of the
len^tli, then (Art. 371.) P,^, is represented by ^'. The
section of rupture in this case is therefore that section in
respect to which i* is first made to satisfy the equation

hi-x^^ — ; or in respect to which the forniula
c,

\

is the least.

If the section of the beam bo uniform, — is constant ; tlie

eertion of rupture is therefore eyidently that which is most
distant from the free extremity of the beam,

413. The beam of GiiEATEsr srsENGTii.

The beam of greatest strength beingthat (Art. 388.)which
presents an equal liability to rupture across every section, or
in respect to -which every section is brought into the state
bordering upon rupture by the same deflecting pressure, is
evidently that by which a given value of Pis made to satisfy
equation (637) tor all the possible values of I, ^„ and tf„ oi'
in respect to which the formula

, Google

62S T

If the beam be nniformly loaded throngliont (Art. 371.)^
tliiB condition becomes

■ (6tl),

or constant, for all points in the length of tho beam.

414, One extkemity of a beam is fiemlt imbedded in
masonkt, and a peessure 19 applied to the otheh
esteemitv in a direction peepehdicular to its length :
to deteemihe the condnioms of the euptttre.

If X represent tho distance of any section of the beam
from the extremity A to which the load P
is applied, and a its whole lengtli, and if the
section of the beam be everywhere the
same, then tlie foiinnla ( 638 ) is least
at the point B, where a; ia greatest : at
this point, therefore, the rapture of the
beam will take place. Eepresenting by
P the pressure necessary to break the
beam, and obserring that in this ease the
]jerpendicnlar npOQ the direction of P
iVora the section of rapture is represented
by a, we have (eq^nation 637)

SI
P=-^ (643).

K the section of the beam be a rec-
tangle, whose breadth is & and its depth e,
then l=i^l<f, c,=-^.

. (643).

K tlie beam be a solid cylinder, whose radius is c, then
(Ai-t. 364.) l=^Trc\ 0,-0.

.•.P=i*S-

. (644).

If the beam be a hollow cylinder, whose radii are r^ and
''t! '^=i*(''i'"~*'/) ; which expression may be put under the
form «CT'(r' + ic) (see Art. 86.), r representing the mean

./Google

THE STEESGTH OF BEAMS.

radina of the hollow cylindei-, and e its thicltneas. Also
(^^f^ (645).

4:15. The strongest form of beam iinder the conditions svp-
poeed in. the last oHiole.

Ist. Let the section of the beam be a
rectangle, and let y be the depth of
tliia rectangle at a point whose distance
from its extremity A is represented by
x^ and let its breadth 5 oe the same
thioughoiit. In this case l=-^y',
c,=i>j; therefore (equation 637) f=

— =iSi— . If, therefore, P be taken

to tepresent the pressure which the
beam is destined just to support, then
tilt- tortn of its section ABO is deter--
mLiipd (Art. 413.) by the equation
6P

^•=ss« <•*«)'

it is therefore a parabola, whose vertex

poi'tion DO of the
A masonry at ei

"I extremity D, its ii

I the same with that of ABO.

2d Let the section be a oitel^, and,
let y represent its radius at distance x
fiom its extremity A, then I=:Jiry',

',= '/, therefore P=^S- so. that the

geometrical form of its longitudina],
section is determined by the oijua-
tion

,m do not rest against
'ery point, but only at its
is form should evidently be

* Thp portion of the bpara imbedded ia llie n
deiiLnbed in Art 417

34

mvy slioiild have the form

, Google

530

■ (6«).

P representing tlio greatest pressure to which it is destined
to be subjected.

il6. The conditions of the etjptuee of a beam suppokteb

AT ONE EXTItEMITT, A^^D LOADED THBOU&HOUT ITS WUOLE
T.ENGJTH,

Kepi e'iCEting tlje weiiiht lestmg upon each incli of its
length a by i^, and observ-
ing that the moment of the
weight upon a length x of
the beam from A, about the
corresponding neutral axis,
K represented (Art. 371.)
by it*ic', it is apparent (Art.
412.) that, if the beam be
of uniform dimensions, ita
section of rupture is BD.
Its strength is determined
by substituting Jjia' forP,^i
respect to (j- ; we thus obtain

(648);

by which equation is determined the uniform load to which
tiie beam may be subjected, on each inch of its length,

For a rectangular beam, whose width is b and its depth
c, this expression becomes

S5c'

. (649).

417. To determine the form of greatest strength (Art. 413.)
in the ease of a beam having .a rectangular section of uni-
form breadth, ■Jiw' must be ^substituted for P^, in equation
(637), and ^\hf for I, and ^ for c, ; whence we obtain by
reduction

y=(a'- («»■

, Google

s Strength of beams.

The form of greatest eti-ength Ib therefore, in this case, the
straight hne joining the points -A and B ; the dietaiice DB
heing detennined by snhstituting the distance AD for x in
the above equation.

That portion BED of the beam which is embedded in the
masonry should evidently be of the same form with DBA.*

418. If, in addition to the uniform load upon the beam, a
given weight W be suspended from A ■^iJ.r' + 'WK must be

^

substituted for P,^, in equation (637) ; we shall thus obtain
lor the equation to the form of greatest strength

■ (651),

which is the equation to an hyperbola having its vertex
at A.t

* It is obTioue that in all eases tho strength of a beam at each point of its
length Is dependeot upon the dimensions of it£ cross section at tliat point, and
ta&t its general form maj in any way be changed without impairing its fllrengtlL
provided those dimensions of the section be CTerywhera preserved.

f Church's Anal. Geom. Art. 124,

, Google

THE 3TKESGTH OF BEAMS.

419. The beam ov geeatest steesgth ih refeeence to the
rokw of its section and to thk vakiation of the
dimensions of iis section, "when supported at onb

The general form of the section must evidently be that
described in Art, 411. Let
the same notation be taken
aa in Art. 365., except that
the depth MQ of the plate
or rib joining the two
flanges is to he represented
by y, and its thickness by e,
so that d,=y, and A,=oy ;
therefore by equation (503),

Also repr^enting by o, the distance of the centre of gravity
of the whole section from the upper surface of tlie beam,
■we hare c,{A, + A^+<yy)={iy + d^oy+(y + d,+id,)A,+id,
A,. Substitating for I and o, in equation (fiSTJ, and for P^, its
value ifw", X being taken to represent the distance AM, and
n the load on each inch of that length, -we have (Art,
413.)

(AX'+A,(7; + cy')(A,+A, + cy)+jl2A,A,+3(A.+ A,)(^j^'
(^+M,)ei/+2(t/+d,+id,)A, + A,d,

(653).

Let the area oy of the section of the rib now be neglected,
as exceedingly small when compared witli the areas of the
sections of the flanges, an hypotliesis which assigns to tlie
beam somewhat le^ than its actual strength; let also tlie
area of the section of the upper flange be assumed equal to
n times that of the lower, or A,=!iAi,
Si^a^ _ (ra + 1) {d,' + nd,') + 12wt/°

" SA,~ '~(2y+d,) + {n + 2)d, '• •''

If the flanges be exceedingly tliin, d^ and d^ are exceed-
ingly small aiidmay be neglected. The equation will then

./Google

THE aTKENGTH OF BEAMS. 533

become that to a parabola wliose vertex is at A and its axig
vertical. Tiiis may therefore be aesamed as a near approxi-
mation to tlie true fonn of tlie carv^e AQC.

Where the material ie cast iron, it appears by Mr. Hodg-
kinson's experiments (Ai-t. 411.) that n is to be taken=6.

SO. A liEAM OF TNIFOIiM SECTION IS eTTPPOETUD AT ITS
EXTEEMiriKS AKU LOADED AT ANT POINT BETWEEN THEM I
IT IS REQUIKED TO DETEEiUUE THE OONDITIONS OF KUFTTJEE.

The point of ruptcre in the case of a imiform section
is evidently (Art. 413.) the point
0, from ■which the load is sus-
pended ; representing AB, AC,
EC, by a, a„ and a, ; and ob-
serving that the pressure P,

upon the point B of the beam = — ^, so that the moment

Wa,a,

(654).

(655);

where W represents the breaking weight, 8 the modulus of
rupture, a the length, b the breadth, c the deptli, and «„ a,
the distances of the point C from the two extremities, all
these dimensions being in inches.

If the load be suspended in the middle, a^=a^=^a,

of P,

, in respect

to the section of ri

have,

by equation (637), '—

_SI

.■.w=^.

If the beam be r,

ictimgular, 1=

iJ,c\

.■.w=l's:?.

. . . .{

. (656).

If the team l)e a soUd cylimUr, whose radiua ^e, then I =
^c\ c^=a ; therefore, equation (654),

■W=lJ— (SST).

,, Google

534 THE STKENGTH OF BEAMS.

If the beam "be a hollow cylinder, whose mean radius ia r,
and its thickness c, I=w(fJ'(r' + i(i'), c^=r-k-\c; therefore,
eqxjation (654),

W=^S'"^V TtV (658).

Iftlie section of the beam be that represented in Art. 411.,
being eveiTwhere of the same dimensions, then, obsei-Ying
that Ao,=-|(^sA, + {:?jA„ nearly, we have, (equations 503 and
654)

I (2A,+A,)a,a,A

. (6S9).

where A„ A, represent the areas of the sections of the upper
and lower flanges, and Aj that of the connecting rib or plate,
and (^1, d^, d, their respective depths.

421. A BEAM IS SUPPORTED AT ITS EXTEEMITIES, AND LOAnED
AT AKY GIVEX FQIHT BETWEEN THEM ; ITB SECTION IB OP A
GIVEN GEOMETRICAL EOKM, BUT OF VARIABLE DIMENSIONS '.
IT IS BEQmEED TO DETERMINE THE LAW OF THIS VARIA-
TION, SO THAT THE STEENGTII OF THE BEAM MAT BE A
MAXIIICM.

W representing the breaking load upon the beam, and
«„ a, the distances of its point
of suspension C, from A and
""i, the pressure P, upon A is

represented by ". If, there-
fore (Art. 388.), X represent
the horizontal distance of any section MQ fi'om the point of
support A, and 1 its moment of inei-tia, and c, the distance
from its centre of gravity to the point where rupture is about
to take place (in this case its lowest point) ; tlien by equa-
tion (637)

W«, SI .„„-,

— -x=— (0G0|.

« e, '

tat. Let the section be rectenywZa?'; let its breadth b be
constant; and let its deptli at the distance w from A be

./Google

THE STKKHGTH OF BEAM8.

represented by y ; tlierefore I^tV^/, Ci=ij'- Si. bstituting
iu tlie above ecLuation and redueing,

^=^r ^^^^^-

The curve AC is therefore a parabola, whose vertex is at
A, and its axis horizontal. . In like manner the curve BC is
a parabola, whose equation is identical with the above, ex-
cept that », is to be aubstitated in it for a,.

2d. Let the section of the beam be a circle. Kepresent-
ing the radius of a section at distance ai from A by y, we
have I=^y', 6^=-y ; therefore by equation (660)

y =

3d. Let the section of the beam be circular ; but let it be
hollow, the thickness of its material being every where the
same, and represented by c. If y= mean radius of cylinder
at distance ic from A, then I=*cy(y' + Ji.'*), c^-=.{y Ar^c) ;

2W», \% + c/

422. Thb beam of g
\ at a given i

Let the section of the beam be that of greatest strength
(Art. ill.). Substituting in equation (660) the value of —
as before in equation (652), and reducing,

Sa te+3^,)i^J'+2(y+4+i<ii)A,+A,rf, " ' '■''*'*'■

If the section ey of the rib be every where exceedingly
small as compared with the sections of the flanges, and if
A,=»Ai,

JW«, (^+l)(4'+W'.) + 12 V ,..r.

Tliere is a value of a; in tliis equation for which y becomes

./Google

536 THE STEEHOTn OF BEAltS.

irayoseible. For values less than this, the condition of uni'
form strength cannot therefore obtain. It 18 only in respect
to those parts of the beam whicli lie between the valnea of
ic (measnred from the two points of support) for which y
thus becomes impossible, tbat the condition of greatest
strength (Art. 388.) is possible. If its proper value be
assigned to n (Art. 411.), this may be assumed as an approxi-
mation to the true form of beam of the &kbatest absolute
STRENGTH. When the material is cast iron, it appears by the
experiments of Mr. Hodgkinson (Art. 411.) that «.=6. A,
represents in all tlie above cases the section of the (netended
flange ; in this case, therefore, it represents the section of
the ^owe?" flange.

The depth CD at the point of suspension may be deter-
mined by substituting a, ior x in equation (665) ; its value is
thus found to be represented by the formula

85=5?-* (666),

433. If instead of the depth of the beam being made to
vai-y so as to adapt itself to the condition (Art. 388.) of uni-
form strength, its breadtli 5 be made thus to vary, the deptli
e reniaining the same ; then, assuming the breadth of the
upper flange at the distance ai from the point of support A
to be represented by y, and the section of the lower flange
to be n times greater than that of the upper; observing,
moreover, that in equation (503) K^^^yd,, A.,=nA.,^nyd[;
neglecting also Aj as exceedingly small when compared with
A, aiid A,, and writing c for d^, we have by reduction,

n+1

Also c, being the distance of the lower surface of the beam
from the common centre of gravity of the sections of the
two flanges, we have Ci(n-|-l)=:c. Eliminating, therefore,
the values of I and o, from equation (660),

"=Wi, I '^'''+^> W+.0§+«rf, ]y (667),

the equation to a straight line. Eacli flange is tlierefore in
this case a quadrilateral figure, whose dimensions ai'C deter-
mined from the greatest breadth ; this last being known, foi

./Google

THE STEENGl'H <

537

the iippei- flange, bv substituting ((, foj- x in the above ec[ua-
tion, and solving in respect of 3/, and for the lower flange
from the equation nbid^=-h,d^, in which &„ 5, represent the
t breadths of the two flanges, and d„ d, tlieir depUis

424. A BEAM K LOADED DNnTOEMLT THKOUGHOUT ITS WHOLU
LENGTH, AND SUPrORTED AT ITS EXTEEMmES: IT 18 KEQUIKED
TO DETEKMINE, 1, The COHDIl'IONS OF ITS KUPTUKE WHEN TIS
OEOSS SEOnOK IS UNIFOSM THEOUGHOUT ; 2. TkE STKONGKBT
FORM OF BEAM HAVING EVEKY WHEEE A EECTANGULAE CROSS

SECTION ; 3. The beam op gkeatiiST strength in refee-

BKOE BOTH TO THE FORM AXD THE VARIATION OF ITS CRCJSS
SECTTON.

1, If the section of the beum be uniform, its point of rup-
ture is determined by foiinula (639)
to be its imdMe point. Eepreeenting,
tlierefore, in this case, the length of
the beam by 2a, the weight on each
inch of its lengtli by c-, and its breadth
by & ; and observing that in this caee
P^,= ij.a'— iij.(is°=^tt', we have by
equation (637)

3S1 laa^.

where (J. represents the load per inch of the length of tlie
beam neceseary to produce rupture. In tlie case of a rectan-
gular beam, this equation becomes

%a^

. (669).

the form of the beam of gi'eatest strength
having a rectangular section of
given breadth 5, let y be taken to
represent its deptli PQ at a point P,
and SB its horizontal distance from
the point A. Tlien I — Jjy',
e,=^; also P^, (equation eST)
representing the moment of the resultant of the pressures
upon AP Sriout the centre of gravity of Y(^=i^ajx,—^v^ \
therefore hj equation (63t) \yaj^—\'^-ii=\^hf ;

./Google

nfi.

tlie er[uation to an ellipse, whose vertex ia in A., and its
centre at 0.

3. To determine the beam of absolute majcimnm strength,
let it bo assumed, as in Art. 422, that the area of the section
of the rib is exceedingly Email as compared with the areaa
<if the sectiouB of the flanges ; and let the ai-ea of the section
of the lower or extended naiige be n times that of the upper ;

■ A_. ,^« I A, \ (71+1) (d'+nd,^) + 12 nyn
then, as m Art. 422, -=-^ i , j . , , ow — — C

also F jy^=i^ax—ifm^ ; whence, by equation (637),

. (670).

+ l){d,'+fid,')+l^f\
2y+d, + {u + 1i)d, f

4. If it be proposed to make the rib or plate uniting the
two flanges everywhere of tlie same depth,* and so to vary
the breadths of the flanges as to give to the beam a uniform
strength at all points under these circumstances ; represent-
ing by y the breadth of the upper flange at a horizontal
distance x from the point of support, we shall obtain, as in
Art. 423,

I

Moreover, F,p,=}i'ax—^x'=itJ-x{2a—x); whence we obtain
by substitution in equation (637), and reduction,

x(^a-x)= (1^) \{n + l) {d,'+nd:)+nno'\y (673) ;

the equation to a parobola,t whose axis is in the horizontal
tine bisecting the flange at right angles, its parameter repre

./Google

THE Sl-EEXGTH OF BEAMS. 539

eented by the coefficient of y in tlie preceding equation, and
half tlie breadth of the flange in tlie middle deteiinined by
the formula

\\n + l) [fi: ^nd^) + \^ntf\M, '^'^'^^

The equation to the lower flange is determined \>j subati-
tuting for y, in equation (673), ~4 ; whence it follows that
the breadth of the lower flange in the middle is equal to
that of the upper multiphed hy the fraction -~ .

fULAR BEAM OF HSIFOKM SECTION, AND UBI-
SOurHOUT ITS LENGTH, IS SUPPOETED BT
TWO PBUPS PLAi tD AT FQUAL DISTAHOKS FEOM 1TB BXTKEM-
ITIEd TO DBTEPMINE THE CONDinONS OF RUPTURE,

It IS evident fiom fuimula (639) that the section of rup-

^ , _ ture of the portion CA of the

~ ^r^^^^^~=^^"^^^^~ beam is at A, and therefore that
— ^-T — — the conditions of its rupture ai"e

t iB"— T,-" I ' ■ t determined (Art, 416,) by the

|1 (I equation

.(676);

where (j-, represents, as before, the
load upon each inch of the length of the beam, b its
breadth, c its depth, and a^ the lengtli of the portion AO.

Again, it is evident that the point of rupture of the por-
tion AB of the beam is at E. Now the value of P,^,
(equation 63T) is, in respect to the portion AE of the beam,
[''a»(a— a,)— if^,a' ; 3a representing the whole lengtli of the
beam [J-, the load upon each inch of the length of the beam
which would produce rupture at E, and merefore v-^a the
resistance of each prop in the state bordering upon rupture ;

also -=^hc'. Whence, by equation (63T), ti'^a{a—a,)~
She'

, (676).

./Google

T!IE STElvKGTI

I' POSITIONS OF THE PEOPS.

If the load f. be imagined to be continually increased, it
is evident that rupture will eventually take place at A or at
E according as the limit represented by eqviation (675), or
iihat represented by equation (676), is fii^st attained, or
according as ii-, or (j., is the less.

Let (J.; be conceived to be trie lees, and let the prop A be
moved nearer to the extremitj' 0; a, being thus diminished,
fj-, "will be increased, and fj-j diminished. Now if, after this
change in the position of the pi'op, ij-, still remains le^ than
fi.„ it IS evident that the beam -will bear a greater load than
it would before, and that when by continually increasing
the load it is brought into the state bordering upon rupture
at A it will not be in the state bordering upon rupture at E,
The beam may therefore be strengthened yet farther by
moving the prop A towards C ; and thns aonUnually, so
that the beam evidently becomes the strongest when the
prop ie moved into such a position that f*; may just equal
p.,. This position is readily detennined from equations (675)
and (676) to be that in which

«,=«( V'3'-l)=-41421S5a (677).

427. A KECTANGULAK BEAM OF UNIFORM SECTION AND UNI-
FOEMLY LOADED IS BtlPPORTEI) AT ITS EXTRFJdrilES, AMD BT
TWO PEOra SITUATED AT EQUAL DI8TANCKS FKOM THEM '. TO
DETERMINE THE CONDITIONS OF EUPTUKE,

Adopting the same notation as in Art. 374., it appears by
. equation (543) that the dis-
tance X, of the point of great-
est curvatm'e of the neutral
line, and therefore of the sec-
tion of rupture in AB from
A (Art. 407.) being that

where -r-^* is the greatest, is

determined by the equation

* The curtaturc of the neutral line being everywhere exceedingly small,

may be aBSumed =1. The expression for the radius of cutraturein terms

tsdf therefore, in this ease, into tli«

./Google

541

H-tCi=P, it being obsei'vecl that, at the section of mixture, tlie
neutral line is concave to tlie axis of a;, and therefore the
second ditferential coefficient (equation 543) negative. The
value of P is that determined by equation (561) ; so that
V+12n'-24n+8

^'=^'^- H2^-3) ^^^^)'

where a represents the distance AE, and na the distance AB.
Let P represent the intersection of the neutral line with
the plane of rapture, and (*, the load per inch of the whole
length of the beam which would produce a rupture at P.
Kow the sum of the moments of tlie forces impressed on
AP (other than the elastic forces on tlie section, of rupture)
is represented in the state bordering upon rupture, by

P,(c,— ^jic,' ; or, since P,=(^iiCi, it is represented by^Pj';

whence it follows by equation (6S7) that the conditions of
tlie raptui-e of the beam between A and B ai-e determmed

by the equation j^P,°=-J-S&c'', or,

=i"-,SJc'' .

. (679).

Eliminating the value of P, between equations (3511 and
(679), we obtain

■W

8w(2w-8)
f 12«,'— 24ft + 8

. (680).

Substituting this value of h-, in equation (C79), and
reducing

_8Sfc-) „(2«-3) 1 .((,31,.

n' + lSn'— 24a.+8

If the points B and 0
coincide, or the beam be
supported by a single prop
in the middle, n=l; there-
fore, by equations (680) and
(681),

(683) ;

, Google

^^ «.

Similai-ly, it appears by equation (547) that tlie point c f
gi'oatest cuiTature between 3i and C is li ; if the rupture (if
tiie beam take place first between these points, it "will there-
fore take place in the middle. Let fij represent the load,
per inch of the lengtli, wliich would produce a rupture at E.
Now, tiie sum of the moments about E of the forces im-
pi-essed upon AE is ~P^a+'P^{a~7m)—ii)-,a'=.(P,+'P^)a—
r,na—i!i.,a'=i^,a'—{(i;a—'P,)na—iii.,a {mice P;+'B,=i'-,a,)=
iKl— 2?i),iJ.,«' + P,«ct. ITierefore by equation {637)
iil-2n)!>-,a' + V,na=iSic' {684).

Substituting tor F, its value from equation (651), and
solving in respect to f*„

.She' I 2)1—3 ) ,,„„,

'---^{^^_4{i-^)4 <^^°'-

K the load be continually increased, the beam will break
between A and B, or between B and 0, according as ('■j
(equation 680) or f*, (equation 685) is the less.

428. ThK BTfiST POSITIONS OF THE PROPS.

It may be shown, as in Art. 436., that the positions in
which the props must be placed so ae to cause the beam to
bear the greatest possible load distributed uniformly over its
whole length, are those by which the values of (*, (equation
680) and fj., (equation 685) are made equal; the former of
these quantities representing the load per inch of the length,
which being uniformly distributed over the whole beam
would just produce rupture between A and B, if it did not
before take place between B and C ; and the latter that
which would, under the same circumstances, produce rup-
ture between B and C if it had not before taken place
between A and B.

Let, then, na represent the distance at which the prop B
must be placed from A to produce this equality ; and let the
value of It-, given by eqiiatioii (679) be substituted for !>■, in
equation (684) ; we shall thus obtain by reduction

> ^ ' 3(l-2>i)- 9{l-2i>)
Solving tills quadratic in respect to P,a,

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The negative sign must be taken in this expression, E-ince
the positive would give P,=(^,a hy equation (6T9), and cor-
responds therefore to the case n=0. Assuniingthe negative
sign, and reducing, we have 3(2n— 1)F,ra=8&(r. Substitut-
ing in this expression for Pj its value from equation (681),
and reducing,

\-V2n^—2in+B

Tlie three roots of this equation are 1-5708T, ■61(yi'8, and
•3699i. The iiret and last are inadmissible ; the one eaiTy-
ing the point B beyond E, and the other assigning to P, a
negative value.* The beet position of the prop is therefore
that which is determined by the value

: -61078 .

429, The conditions of toe euptube of a eectahgulae

BEAM LOADliD UNIFORMLY THROUGHODT trS LENGTH, AITD
HAVING ITS ESTKEMITIES I'ROLONOHD AND FISIILY IMBEDDED
m MASONEY.

It Jias been shown (Art. 376.) that the conditions of the
deflexion of the beam are, in this case, the same as though
its extremities, having been prolonged to a point A (see j£g.
p. 540.), such that AB might equal -GSOSAE, bad been sup-
ported by a prop at E, and by the resistance of any fixed
surface at A. The load which would produce the rupture
of the beam is therefore, in this case, the same as that which
VTOuld pi-oduce the rupture of a beam supported by props
(Art, i27.) between the props, and is determined by that
value of f*j (equation 685) which is given by the value '6202
of n. It is, however, to be observed that the symbol a

" We may, nevertheless, anpposo the eitramity A, instead of being sup
porWd from beneatli, to be rlnned down by a resistance or a pressure acticg
from above. This cnse may ooeiir in practice, and the best position of the
props corresponding to it is that which is determined by the least root of the
equalion, viz. ■26994.

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TEE BTEENGTH OF COLOMNB.

represents in tliat equation
the distance AE {fig. Art.
427.) ; and tliat if we take
it to I'epvesent the distance
BE in that or the accompa-
nying lignre, we mnst sub-
stitute z ' for a in equa-

tion (685), since «=BE=AE-AB=(l-7i.)AE ; «o that
AE=-— -. This substitution being made, equation (685),

becomes

_JiU (2^-^3)(l-^t)'
^''^-^ a' n'~4(l-w)- '

and snbstitiiting the yalue ■
tion

Bbc'

i for n, we obtain by reduc-

■ (<58^),

by which fonnula the load per inch of the length of the
beam necessary to produce mpture is determined.

If the beam had not been prolonged beyond the points of
support B and 0 and inabedded in the masonry, then the
load per inch of the length necessary to produce rnptnre
would have been represented by equation (669) : eliminat-
ing between that equation and equation (687), we obtain
[i.,=3f* ; so that the load per inch of the length necessary to
pi'oduce rapture is 3 times as great, when t}ie extremities
of the beam are prolonged and hrmly imbedded in the ma-
sonry, as when they are free ; i, e. the strejigth of t}te iea/m
is 8 iwies as greed in, tlte one case as m, the ot/ier.

430. Toe steengh or

Eor all the knowledge of this subject on which any reli-
ance can be placed by the engineer he ia endebted to expe-
riment.*

* The hypothesifl upon whioli it has bean customary to found the theocatioal
discussion of it, is so obriouely IneufEcient, and tbe results huTe been shown
by Mr. Hodgkineon to be bo little in aooordanoe with those of practice, that
the high sanction it has received from kbours such as those of Euler, Lagrange,
Poiason, and Navlei", can no longer establish for it a clHim to be admitted
among the concluaons of science. (See Appendix K.)

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■rilJ! STBENGTH OF COLUMSS.

The following are the principal results obtained in tlie
valuable series of experimental inqniriee recently instituted
by Mr. Eaton Hodgkiuson.*

FOKMULiB BEPEKSEHTIHa THE ABSOLUTE 8TEESGTH OT' A CYL-
nrOKIOAL COLIJMH TO SUSTADT A PEESSUKE IS THE OTEECTIOU
OF ITS LBMGTH.

D=external diameter or side of the scLiiare of the column
in inches.

D,=iiiternal diameter of hollow cylinder in inches.

L=leiigth in feet,

"W ^breaking weight ia tons.

«......„.o......

the Oolmnn esceediiig

BdUi Ends beiii^ S^t, Ifae

Lsnelhottlie Column
exceeding lliirtj tJines ila

Hollow ojlindrical column of (

Solid oylindricnl column off
■wrought iron - - f

SoUd square pilkr of Dautzic )
oak (dry) . . - - f

Solid square pyiar of red deal )
(dry) f

ir-i«"""

, In all eases the strength of a column, one- of whose ends
■was rounded and the other flat, was found to be an arith-
metic mean between the strengths of two other columns of
the same dimensions, one having both ends rounded and the
oiier having both ends flat.

The above results only apply to the ease in which the
length of the column is so gi'eat that its fractuTe is produced
wholly by the bending of its material ; this Kmit is fixed by
Kr. Hodgkinson in respect to columns of cast iron at about
fifteen times the diameter when the extremities are rounded,

* From a paper by Mr. Hodgkinson, publlabed in the second part of the
Transactions of the Royal Society for 1840, to which the royal medal of the
Society was awarded. The eiperimenta were made at the expense of
Mr. Fairbaim of Manchester, by whose liberal encouragement the researches
of practical science have been in other respects so greatly adyaiiced.

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546 TOESION.

and thirty times the diameter when they are flat. In
shorter columns fracture takes place partly by tlie crushing
and partly by the bending of the material. To theee shorter
columns the following nile was found to apply with sof-
ficient accuracy : — " It' W, represent the weight in tons
■which would break the column by bending alone (or if it
did not crush) aa given by the preceding formula, and "Wj
the weight in tons which would break tlie column by crush-
ing alone (or if it did not bend) as deteiTained from the
preceding table, then the actual breaking weight W of the
column is represented in tons by the formula

^=w:tiw; ("'**'>

Oolwrri/ns enlari'ged in the middle. It was foiind tliat the
etrengths of columns of east iron, whose diametera were from
one and a half times to twice as great in the middle as at
tlie extremities, were stronger by one seventh than solid
columns, containing the same quantity of iron and of the
same length, when their extremities were rounded ; and
stronger By one eighth or one ninth when their extrei
were flat and rendered immoveable by discs.

431. Eelative strength of long columns of cast ntow,

WROUGHT mON, STEEL, AND TBIBEE OF THE SAME DIMENSIONS.

— Calling the strength of the cast iron column 1000, the
strength of tlie wrought iron column wi ll, according to these
expenments, be 1745, that of the cast steel column 2518, of
the column of Dantzic oak 108'8, and of the column of red
deal T8-5.

Effect of drying on the strength of columns of ti'mJ>er. —
It results fi'om these experiments, that the strengtli of short
columns of wet timber to resist crushing is not one kalfihAt
of columns of the same dimensions of dry timber.

Torsion.

432. The elasticity of torsion.

Let ABCB represent a solid cylinder, one of whose trane

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547

• I

''ff

p Terse sections AEB is immoveably fixed,

'°~'~^ lud every otlier displaced in its own plane,

nibout its centre, by tbe action of a pres-
*" sure P applied, at a given distance a trom
the axis, to the section CD of the cylinder
m the plane of that section and round its
centre ; the cylindei' is said, nnder these
circumatanees, to be subjected to torsion,
and the forces opposed to the alteration of
if* form, and to its ruptnre, constitute its
lesietance to torsion.

Let aa.h^ be any section of the cylinder
i^hose distance tronl tlie section AEB ia
1 presented by x, and let a/3 represent that
liaineter of the section aaJ/3 which was
} ii ill I II imeter AB before the torsion commenced;
Jet 'ih he tlie j rejection of the diameter AE npon tbe sec-
tion aahfi, and let the angle aea be represented by i.

Now tlie elastic forces called into action npon the section
oab^ are in ec[ailibrium with tbe pressure P. But these
elastic forces result from the diaplaoetnent of tbe section
«aJ,3 upon its immediately subjacent section. Moreover,
the actual displacement of any small element aK of the
section aalb^, upon the subjacent section, evidently depends
partly npon tbe angvlai displacement of tbe one section
upon the otbei, and partly npon the distance p of the
element m question fiom the axis of the cylinder. Now the
angle aca oi ^ is evidently tbe sum of the angular displace-
ments of all the sections between cuiW and AEB upon their
subjacent sectionh , ai d the angular displacement of each
upon its subjacent section is the same, the circumstances
•iffecting tbe displacement of each being obvioialy the same :
iKo the numbei ot these sections varies as a;, and the sum
of their angulii displacements is represented by ^ ; there-
foie the anguHi di'^;placement of each section upon its sub-

jacent section V

-, and tbe actual displacement of

the smiU element aK ot the section m.h^.v

Now

tlie raateud beiui, elastic, the pressure which must be
ipplied to this <>lement m order to keep it in this state of
displacement \ itieo as the amount of the displacement

(Art, 345.), or as -p. Let its actual amount, when referred

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to a unit of Burface, lae represented by G-p, where Q- is a

certain constant dependant for its amount on ttie elastic
qualities of the material, and called the modulus of torsion ;
then will the force of torsion required to keep the element

£iK in its state of displacement be represented by G-pAK, and

6
its moment about the axis of the cylinder by G-p'AK. So

that tlie snni of the moments of all snch forces of torsion in
respect to the whole section Oab^ will be represented by

G-2p°AK, or by G-I, if I represent the moment of inertia

of the section aboat the axis of the cylinder. iN'ow these
forces are in equilibrium with P ; therefore, by the principle
of the equality of moments,

P«=:GI-

If r represent the radius of the cylinder, l=^r' (Art.
85.), Substituting this value, representing by L the whole
length of the cylinder, and by & the angle through which
its extreme section CD is displaced or through which OP is
made to revolve, called the oftffle of torsion, and solving in
respect to ®,

-(3- ?••■■«■

Thus, then, it appears that when the dimensions of the

T cylinder are given, the angle of torsion © varies

directly as the pressure P by which the torsion
is produced ; whence, also, it follows (Art. 97.)
that if the cylinder, after having been deflected
through any distance, be set free, it will oscil-
late isoehronously about is position of repose,
the time T of each oscillation being represented
in secoiids (equation 76) by the formula

ance by equation (690) P= I ^-^^((Sa); in which exprefsi

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T0E8I0S. 546

(0») represents the length of the path described by the
point P from its position of repoBe, so that the morlng force
upon the point P, when the pressure prdacing toi-eion is
removed, varies as the path described by it from its position
of repose.

The above is manifestly the theory of Coulomb's Torsion
Balance* W represents in the formula the weight of tht
mass supposed to be carried round by the point P, and tht
inertia of the cylinder itself is neglected as exceedingly
email when compared with the inertia of this weight.

The torsion of rectangular prisma has been made the sub-
ject of the profound investigations of MM. Oaucbyf , Lame,
et Clapeyront, and Poi^on.| It results from these investi-
gations I that if i and o be taken to represent the sides of the
rectangular section of the prism, and the same notation be
adopted in other respects as before, then

e=3PL»(SN^ (S92).

M. Canchy has shown the values of the constant G to
be related to those of the modulus of elasticity E by the
formula

G=aE (693).

Li using the values of G deduced by this formula from
the table of moduli of elasticity, all the dimensions must be
taken in inches, and the weights in pounds.

433, Elasticity of torsion in a solid having a cieculap
secnos of variable diilessions.

Let ab represent an element of the solid contained by

* Hlvietratlons of Mechaiiice, Art. ST.

■| Exereiiies de Math^matiques, i' annce.

I Crelle's Journal. g Memoirea de I'Acaclemie, tome viii.

I Navier, Resume dea I.cqoos, iic, Art. 159.

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>5'

planes, perpendiculai' to the axis, whose die

tance fiom one another ie represented by

tile exceedingly small increment ^a; of the

distance 'b of the section ab from the fixed

section AB, and let its radius be repre-

bented by y ; and suppose the whole of the

boli<l except this single element to become

iij:id, a supposition bywhich the conditions

of the equiUbrium of this particular element

ill lemain unchanged, the pressure P re-

I iinmg the same, and being that which

I ( iduces the toreion of tliis single element.

- -^ U hence, rejpresenting by Afl the angle of

toiBion of this element, and considering it

(, <"\hn<lei whosf lengtli is ^x, we have by equation (689),

Mibbtitutmg for I ito "v alue t^/,

Passing to the limit, and integi'ating between the hmitfi 0
and L, observing that at the former limit S—0, and at the
latter i=®.

2Pa r^
'-^gJ y' ■

. {&M.)

If the sides AC and BD of the solid be straight lines, its
form being that of a truncated cone, and if r, and r, repre-
sent its diameters AB and CD respectively ; then

dx _ L

d]/~ »',— ^j
Also,

L n n

. . (695).

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434, Teee kuptckb of a oyujsdeb bt tokston.

It is evident that rapture will first take place in respect
to tiicffle elements of the cylinder which are nearest to its
Birrface, the displacement of each section upon its subjacent
section being greatest about those points which are nearest
to its circumference. If, therefore, we represent by T the
pressure per square inch which will cause mpture by the
Bliding of any section of the mass upon its contiguous sec-
tion,* then will T represent the resistance of torsion per
square inch of the section, at the distance r from the axis, at
the instant when rupture is upon the point of taking place,
the radius of the cylinder being represented by r. Whence
it follows that the displacemont, and therefore the resistance
to torsion per square inch of the section, at any other dis-
tance p from the axis, will be represented at that distance by
-L the resistance upon any element aK, by - pAK, and the
sum of the monients about the axis, of the resistance of all
such elements, by _ Sp^AK, or by - I, er substituting for I

its value (equation 64) by iTT/. But thes

in equilibrium with the pressure P, which produces torsion,

acting at the distance « from the axis ;

.-. Pa=iT*i-' .... (696).

It results from the researches of M. Cauehy, before referred
to, that in the case of a rectangular section whose aides are
represented by i and c, the conditions of rupture are deter-
mined by the equation

The length of a prism subjected to toreion does not affect
the actual amount of the pressure required to produce rup-
ture, but only the angle of toi-sion (equation 690) which
3 rupture, and therefore the space through which

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the pressure must be made to act, and the amount of wokr
■wMch must ie done to produce rupture.

According to M. Cauchj, the modulus of rupture by tor-
sion T is connected with that S of rupture by tranaverse
strain by the ec[uation

T=4S (698).

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P^KT VI.

IMPACT*

435. The impact of two bodies whme centres of gsavits
mote m the same eight lihe, and -whose point of cok-

TACrr IB IN THAT LIKE.

From the period when a body first receives the impact of
another, until that period of the impact when both more for
an instant with the same velocity, it is evident that the em--
faces must have been ia a state of continually increasing
compresBion : the instant when they acquire a common velo-
city is, therefore, that of their greatest compreBsion. When
this common velocity ia attained, their mutual pressures will
have ceased if they be inelastic bodies, and they will move
with a common motion ; if they be elastic, their surfaces
will, in the act of recovering their forms, be mutually
repelled, and the velocities will, after the impact, be dif-
ferent from one another.

436. A noDY whose ^veight is "W,, axd which is moving

IS A HOEiaONTAL DIRECTION WITH A UBIFOEII VELOCITY
ntEPEESENTISD BY V,, IB IMPINGED UPON BY A SECOND BODY
WHOSE WEIGHT IS W,, AND WHICH IS MOVING IN THE SAME
STEAIGHT LINE WITH THE VELOCITY V, ; IT 18 EEQUIEED TO
DETEKillNE THEIR COMMON VELOCITY Y AT THE INSTANT OF
GEEATEST CX)MPEESStON.

Let/, represent the decrement per second of the velocity
of "W"i at any instant of the impact (Art. 94.), or rather the
decrement per second which its velocity would experience
if ihe retarding pr^svire were to remain constant ; then wilJ

« Note [v), Ed, App.

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w

— \fi represent (Art. 95.) the effective force upon "Wj ; and if
_/", be taken to represent, under the eame circmnstancee, tlie
increment of velocity received by W„ then will — ^/j repre-
sent the effective force upon "W",. Whence it follows, by the
principle of D'Alembert (Ai't. 103.), that if these effective
forces be conceived to be applied to the bodies in directions
apposite to those in which the corresponding retardation
and acceleration take place, they will he in equilibrium with
the other forces apphed to the bodies. Bnfc, by supposition,
no other forces than these ai'6 appHed to the bodies : these
are therefore in eCLuilibrium with one another,

Let now an exceedingly small increment of the time from
the commencement of the impact be represented by '^t, and
let ^v, and Au^ represent the decrement and increment of
the velocities of the bodies respectively during that time,

.-.(Al-t. 95.)/A(=A-i)„/,.V=Ai,^;

.'. (equation 699) W, . Av^—W^ . a^^ ;

and this equality obtaining throughout that period of the
impact which precedes the period of greatest compression, it
follows that when the bodies are moving in the sarne direc-
tion

W,(V,-V)="W",(V-T,) (700) ;

since Y,— V represents the whole velocity lost by W, during
that period, and V— T, the whole velocity gained by W,.

If the bodies be moving in opposite direccions, and their
common motion at the instant otgreatest compression be in
the direction of the motion of W^, then is the velocity losl
by W, represented as before by (V,— V); but the sum of
the decrements and increments of velocity communicated to
"W",, in order that its velocity Y^ may in the first place be
destroyed, and then the velocity Y communicated to it in aa
opposite direction, is represented by (Yj-I-Y).

.■.W.(Y,-Y)=WXY,+V). .
Solving these equations in respect to Y, we obtain

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^= wfw. ■ '™i)^

the sign ± being taken according as tlie motions of tin
bodies before impact are both in the same direction or in
opposite d ire cti Otis.
If tlie second body was at rest before impact, V^=0, and

^=w;w, (™^)-

If the bodies be equal in weight,

The demonstration of this proposition is wholly indepen-
dent of any hypothesis as to the nature of the impinging
bodies or their elastic properties ; the proposition ie there-
fore true of all bod'es, whatever may be their degrees of
hardness or their elasticity, provided only that at tfie
instant of greatest compi^ession every part of each body
partakes in the common velocities of the bodies, there being
no relative or vibratory motion of the parts of either body
among themselves.

437. To DETEKJtINE THE WOKK EXPENDED UPON rKODDCINO
THE STATE OF THB GKEATEST COIII'KESSION OE THE SUK-
FACEB OF THE BODIBS.

The same notation being taken as before, the whole work
accumulated in the bodies, before impact, is represented

W W

bv i — 'V,'-l--fc— ^V.' : and the work aecumniated in tliem

at tlie period of gi-eatest compression, when they move with

the common velocity Y, is represented by i — "V'.

Now the difference between the amounts of work accumu-
lated in the bodies in these two states of their motion has
been expended in producing their compression ; if, there-
fore, the amount of work thus expended bo represented by
«., we have

"W" "W W +'W

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or substituting for V its value from ecLuation CTOI), and
reducing,

1/ WAV \

This expression represents tlie amount of work permcmently
lost in the impact of two inelastic bodies, their common
velocity after impact being represented by eqiiation (701).
If W, be exceedingly great as compared witli w „

»=?'(V,TV-,)' .... (704).

438. Two ELASTIC BODIES IMPINGE UPON ONE AXOTIIEIt ; IT IS
EEQDIKKD TO DETERMINE THE VELOCITY AFTEK IMPACr.

If tlie impinging bodies be perfectly elastic, it is evident
that after the period of tlieir greatest compression is p^sed,
they -will, in the act of expanding tlxeir surfaces, exert
mutual pressures upon one another, ■which are, in corres-
ponding positions of the surfaces, precisely the same with
those which they sustained whilst in the act of compression ;
whence it follows tliat the decrements of velocity expe-
rienced by that body whose motion is retarded by this
expansion of the surfaces, and the increments acquired by
that whose velocity is accelerated, will be equal to tliose
before reeeivefl in passing flirongh correspondmg positions,
and therefore the whole decrements and increments thus
received dm'ing the whole expansion equal to those received
during the whole compression.

Now tlie velocity lost by W, during the compressiou is
represented by ("V",— T); that lost by it during tlie expan-
sion, or from the period of greatest compression to that
when the bodies separate from one another, is therefore
represented by the same quantity, But at the instant of
gi'oatest compression both bodies had the velocity V ; the
velocity v, of Wj at the inst^mt of separation is therefore
V— (Vj— V), or 3V— V,. In like manner, the velocity
gained by W, during compression, and therefore during
expansion, being represented by (VtVj), and its velocity
at the instant ol greatest compression by V, its velocity v^
at the instant of separation is represented by 'V"-i-(Y^V,),
or by 2V=pVj, the sign =F being taken according as the

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ir.otioTi of the bodies before impact was in the same of
opposite directions,

Substitntiiig for V its value in these expressions (equation
701), and reducing, we obtain

W.-i-W, ■■■■\"^-^)

If the bodies be perfectly elastic and equal in weight,
u,— Y„ !)5=Vi ; they therefore, in tliie case, mterchcmm their
velociries by impact ; and if either was at rest before impact,
the otJier will be at rest after impact.

If Wj be exceedingly great as compared with W„ ^,=
— V,±2Vj, w,= ±V5. In this case ^i is negative, or the
motion of the lesser body alters its direction after impact,
when tlieir motions before impact were in opposite direc-
tions ; or when they were in tne same direction, provided
that 2Y, be not greater than V,.

439. If the elasticities of the balls be imperfect, the force
with which they tend to sepai-ate at any given point of the
expansion is different from that at the corresponding point
of the compression ; the decrements and increments ot the
velocities, produced during given corresponding periods of
the compression and expansion, are therefore different;
whence it follows that the whole amounts of velocity, lost
by the one and gained by the other during the two periods,
are different : let them beai- to one another the ratio of 1 to e.
Now the velocity lost during compression by "W, is under
all circamstancea repr^ented by (V,— V) ; that lost during
expansion is therefore represented, in this case, bye (yi"~^i
thei-efore, «,=V— e(V",~V)=(l + e)V— eV,. In like man-
ner, tlie velocity gained by "W, during compression is in all
cases represented ny (VT V j) ; tliat gained during expansion
is thei'efore represented by e{YT v,); therefore, «,=V-|-
,i(V:^V,)=(l-he)V=FeVa- Substituting for V, and reducing,

_j-(-W^_g-W,)Y,+(l + g)"W,V.

. (708).

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440. In the impact of two elastic bodies, to deteeminb
the aoghmulated wokk, ok one half the vis viva, lost
by the one and gained by the other.

The vie viva lost "by "W, during the impact is evidently

represented by — -\ , -v, =— i(v, —v,) = —^l V, —

Y'\ =^il + e){Y,-Y){YXl-e) + Y{l + e)l.

Snbetitnting in thia expression its value for V (eq^iiation
701) reducing and representing by i*, one half the vis viva
lost by W, in its impact, or the amount by which its accumu-
lated work is diminished by the impact (Art. 67.),

_ (i+.)WW.(y,^T,)

(l+e)\V,V,i .... (709).

Similarly, if u, be taken to represent one half the vie viva
gained by W„ or the amount by which its accumulated work
IB increased by the impact, then

(l + e)W,V.} (710).

441. Let u he taken to represent the whole amount of the
work accumulated in the two bodies before their impact,
which is lost daring their impact. This amount of work is
evidently equal to the difference between tliat gained by the
one body and lost by the other; so tliat u=-u,~u,. Substi-
tuting the values of u^ and u, from the preceding equations^
and reducing, we obtain

_(l-e')W,W,(Y,TY,)'

• (I'll).

Tills expre^ion is equal to one half the vis viva lost dui-ing
the impact of the bodies. If the bodies be perfectly elastic.
6=1, and ii:=(i. In this case there is no real loss of vis viva

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in tbe impact, all tliat which the one body yields, during the
impact, bebig taken up by the other.*

442. In tlie preceding propositions it has be
that the motions of the impinging body, and the body ii
pinged upon, are opposed by no resistance whatever during
the period of the impact. There is no practical case in
■which this condition obtains accm-ately. If, nevertheless,
the resistance opposed to the motion of each body be small,
as compared with the pressiu'e exerted by each upon the
other, at any peiiod of the impact, then it is evident that all
the circumstances of the impact as it proceeds, and the mo-
tion of each body at the instant when it ceases, will be very
nearly the same as though no reeistance wei'e opposed to the
motion of either.f

443. As an illustration of the pi'inciple established in the
last article, let it be required to determine the space through

* It haa baen customary, iievertlieless, to apeak of a loss of via Viva ia the
impact of perfectly elastic bcdiea. This loss is in. all such cases to be under-
etood only as b. loss expevieiieed by atie of the todies, and not as an absolute
loss. When the impinging bodies are perfectly elastic, it is evident that the
one fliea away with all the via viva which ie lost in the impact by the other.

f Let Pi and Pa repreaont reaiatancea oppoeed to the motions of two im-

pioging bodies whose weights ace Wi and Wa ; alao let — /i, md - — /a re-

38 at any period of tiie iinptLCt ;

j; by t the time occupied in the impact, up to the period of
greatest compression, by V Ibeic common velocity at that period, and by vi
and «s their velocities at any period of the impact, and substituting for /i and
/a their values (equation 12),

--§-■•■=»•

n the limits 0 and i,

7' ''■"'''=7' lT-T,H-/(P,+P.).».

Now if Pi and Pa be not esceedingly great, the integral in the second niemhei
of the equation is exceedingly small as compared with its other terms, and may

bo neglected ; the above equation will ti^en become identiciil with equation

(TOO).

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wliicli a nail will be driven "by the blow of a hammer ; and
let it be supposed that the resistance oppoaed-to the driving
of lie nail is partly a constant resistance overcome at its
point, and partly a resietanee opposed by the friction of the
mass into which it is driven upon its sides, vaiying in amount
directly with the length of it x, at any time imbedded in the
wood. Let tliis resistance be represented by a+(3at; then
will the work which must be expended in driving it to a
depth I) be represented (Art, 51.) by

f{cL + ex)d», or by (aD+iSB').

Let "Wj represent the weight of the nail, and V the velocity
with which a hammer whose weight is W, must impinge
upon it to drive it to this depth, and let the surfaces of the
nail and hammer both be supposed inelastic ; then will the
wort accumulated in the hammer before impact be repre-

W
sented by i — 'Y", and the amount of this work lost during

the impact by the compression of the surfaces of contact will

1 / WW \
be represented (equation 711) by g-l-^' ' -^| V. The work

remaning, and effective to drive the nail, is therefore repre-
sented by the difference of these quantities ; and this work
being tliat actually expended in diiving the nail, we have

-;^.=''""+'™' ■ • ■ ■ <"''■

by the solution of which qnadratie eq^uatioi),Dmay be deter-
mined.

444. Two SOLID PlilSMS HAVE A COMMON AXIS; THE EXTSEM-
ITT OF ONE OF THEM RESTS AGAINST A FIXED 3UEFACE, AND
ITS OPPOSITE EXTREMITY BEOEIVES TBE IMPACT, IN A HORI-
ZONTAL DIRECTION, OF THE OTHER PRISM: IT IS KEQUIllED
TO DETFJIMINE THE COMPRESSION OF EACH PRISM, THE LIMITS
OF PEKFECT ELASTIOITI NOT BEING PASSED IN THE IMPACT.

Let W represent the weight of the impinging prism, and

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V its velocity before impact ; L^ and Lj tlie lengths of the
prisms before compression ; E, and E, their moduli of elas-
ticity ; K, and K, their sections ; ?, and l^ tlxe greatest com-
pressions produced in them respectively by the impact ;
then will the amoonte of work which must have been done
upon the pmme to produce these compressions be repre-
sented {equation (486) by the formnlEe

1 the whole work thus expended by

But this work has been done by the work A — V, accumn-

lated (Art. 66) before impact in the impinging body, and
that work has been exhausted in doing it ;

Moreover, the mutual pressures upon the suifaces of con-
tact are at every period of the impact equal, and at the
instant of greatest compression they are represented respeo-

tively (eqiiation 485) by — y— -and — t — ;

.KM^KAi^p (713).

Eliminating ?, between this equation and the preceding, and
reducing,

'•-ii\[h^iky"^^ ('")=

in which expressions l, represents the greatest compression

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of the prism wliose section is K„ and P the driving pressure
at the instant of gi'eatest coinpreaeion.

445. The mviual pressures F of the surfaces of contact at
miy period of the impact.

Let I represent the space described by that extremity of
the impinging prism, by which it does not impinge : it is
evident tliat mis space is made up of the two corresponding
compreasions of the surfaces of impact of the prisma ; bo that
if tliese be represented by Z, and l„ then 1=1^+1,. But

(equation 713) ^i=tfV' ^^^'iTp' therefore l = 'F[^^-f-

ii .

446. A measure of the compressibility of tlie prisms.

If X be taken to represent tlie space through which that
■extremity of the impinging priam by which it does n(<t
impinge will have moved when the mutual pressure of the
sm€aces of contact is 1 lb, ; or, in other woi-de, if ^ repre-
sent the aggregate space througb whicli the prisms would
be compressed l)y a pressure of 1 lb. ; then, by the preced-
ing equation,

X may be taken as a measwre of the agyremta campremr
hiUfiy of t'he prisms^ being the space through wMch their
opposite eietremtties woidahe made to approach one another
! .qf 1 lb. applied in the di/reotion of their

M \ and X, represent the spaces through which the
prisms would s&oerally be compressed by pressures of 1 lb.

applied to each, then X,=|-~-, \=^^ ; therefore X=\4-

,„ or the aggregate compressibility of the two prisms ie
ei^nal to the sum of their separate compressibilitiea.

./Google

M7. The work u expended upon i/ie eompressitm of the
prisms at any period of the impact.

The work expended upon tbe compression ^, is repre-

KE

sented \>j ^ -r—^l' ; or substitnting its Talue for Z, (equation

Y13), it is represented by ^^=-—'2'. And, similarly, the work

expended on the compression I, is represented by iTr4i"I"j

therefore ^6=41:f^ +^^4; |P'; or siibstitnting for P its
value from equation (710),

-=»1o;+ia.)~'=ix m-

as. The iielooity of the impinging hody at ami/ period of the
impact, the impaet leing supposed to take place vertically/.

It is evident that at any period of the impact, when the
velocity of the impin^ng body is represented by «, there

will have been expended, upon the compression of the two
bodies, an amount of work which ia represented by '^
: aceamulated in the impinging body before imp;

increased by the work done upon it by gravity during the
impact, and diminished by that which still remains accu-

mulated in it, orbyi^ — V+W^— ^ — v".

Eepresenting, therefore, by u the work expended upon

the compression of the bodies, we have i — ■Y'' + 'Wl~

Substifutine, therefore, for u its value from equation

(ns),

, Google

:=T-«;,4(k^ + K^,)

. (719).

Oi Bubstituting for I its value ii

terms of P (equation 716),

. (Y20).

The Pile Drivek,

It is evident that the pile will not begin to be
driven until a period of the impact is at-
tained, when the pressure of tlie ram upon
its head, together with the weight of the
pile, exceeds the resistance opposed to its
motion by tlie coherence and tlie friction of
the mass into which it is driven. Let this
resistance be represented bj P ; let V repre-
sent the velocity of the ram at tlie instant of
impact, and v its velocity at the instant when
the pile begins to move, and W„ W, tlie
weights of the ram and pile ; then, since the
pile will have been at rest during the whole
of the intervening period of the impact, since
moreover, the mutual pressures Q of the sur-
faces of contact are at tlie instant of motion
represented by P— "W,, we have by equation
(720)

If the value of v determined hy tliis equation be not a
possible quantity, no motion can be communicated to the
pile by the impact of the ram ; the following inequality is
therefore a condition necessary to the diiving of the pile,

After the pile has moved through any given distance, one
portion of the work accumulated in the ram before its
impact will have been expended in overcoming, through
that distance, the resistance opposed to the motion of the

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IMPACT. 505

pile; anotlier portion will have been expended upon the
eompreseion of the surfaces of the ram and pile ; and the
i-einainder will be accumnlated in the moving masses of the
ram and pile. The motion of the pile cannot cease until
after the period of tlie greatest compression of the raia and
pile is attained ; since the reaction of the surface of the pile
upon the ram, and therefore the driving pressure upon the
pile, increases continnally with the compression. If tlie
surfaces be inelastic, having no tendency to recover the
forms they may have received at the instant of greatest
compression, they will move on afterwards with a common
velocity, and come to rest together ; so that the whole work
expended pi'ejudicially dunng the impact will be that
expended upon the compression of the inelastic surfaces of
the ram and pile. If, however, both surfaces be elastic,
that of the ram will return from its position of greatest
compression, and the ram will thus acqmi'e a velocity rela-
tively to the pile, in a direction opposite to the motion of
the pile. Until it has thus reached tlie position in respect to
the pile in which it first began to drive it, their mutual
reaction Q will exceed tlie resistance P, and the pile will
continue to be driven. After the ram has, in its return,
passed this point, the pile will still continue to be driven
through a certain space, by the work which has been accu-
mulating in it during the period in which Q has been in
excess ot P. When the moti(/n of tirn pile ceas^, the ram
on its return will thus have passed the point at which it
first began to drive the pile : if it has not also then passed
the point at which ita weight is just balanced by the elas
tieity of the surfaces, it willhave been continually acq^u'rin t
velocity relatively to the pile from the period f at t
compression ; it wiU thus have a certain veloc ty nd a
certain amount of work wdl be accumulated in t hen t! e
motion of the pile ceases : this amount of work tOp, the
with that which must have been done to produce that con
pression which the surfaces of contact retain at that t nt
will in no respect have contributed to the driv n t the
pile, and will have been expended uselessly. If th
its return has, at tie instant when the motion f tl p le
ceases, passed the point at which its weight would just be
balanced by the elasticity of the surfaces of contact, ita
velocity relatively to the pile wUl be in the act of diminish-
ing; or it may, for an instant, cease at the instant when the
pile ceases to move. In this last case, the pile and ram, for
an instant, coming to rest together, the whole work accuiDu-

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B66 IMPAOT.

lated in tlie impinging ram will have been usefully expeiideij
iu driving the pile, excepting only that by which the remain-
ing compression of the sm-facea has been produced ; which
compressiou is less than that due to the weight of the rara.
This, thei-efore, may be considered the case in which a maxi-
mum useful effect is produced by the ram. The following
article contains an analytical discussion of these conditions
under their most general form.

450. A pi-ism impinged upon i^
di/feohon of its axis, and its motion is o^osed hy a eon-
sta/nt presswe P ; it is required to determine the con-
ditions of the motion dwn/ng the period of impaot, the
oimmmstances of the impact ieinff in other respects the
same as in Ai'tiele 448.

Let f^ "indy", lepresent the additional velocities which
wiuld be lobt and acquiied per second (see Art.
y^l hi tiie impinging prism and the priem
pinged upon, if the pressures, at any instant
I eiAtmg upon tliem, were to remain from that

stant constant , then will — f, -^, represent

tl e efFeetive forces upon the two bodies (Art, 103)
nr the piesbuies whicli would, by the principle of
BAlembeit, be m equilibrium with the unbal-
anced preBsnies upon them, if applied in opposite
directions
Kow the unbalanced pressure upon the system

i| " BP composed of the two prisms is represented by

' ■ (W,+W,-P),

. "^L/+ ^y-W -|-W,~P (733) ;

also the unbalanced pressure upon the prism PQ=W„-i-
Q — P, where Q represents the mutual pressure of the prisms
3tQ;

"W"

.-.— y;=Av,+Q-p (724).

Let A have been the position of the extremity B of the
impinging prism at the instant of impact ; and let (b, repre-
sent the apace through which tlie aggi-egate length EP of
the two prisms has been diminished since that period of the

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IMPACT,

567

impact, and iK, the space through which tlie point F ]
moved ; then (eqnation tl(>)

'\K,E/ K,eJ

. {m).

Also AB~!e,+iii,; therefore velocity of point B:= — "7?/"^'

(Art. 96); therefore /..=§+5=§+/..

Substituting these values of y, and Q in equations (T23) and
(724), and eliminating/, between the resulting equations,

Integrating tliie equiition by the known niles,t we obtain

x^=A. sin. yt+~B cos. yt+^Sr {'i'ST);

in which expression the value of / is determined by the
equation

and A and B are certain constants to be detennined by the
conditions of the question. Substitntina; in equation (724)
the value of Q from equation (725), and solving in respect
to/,,

/.=^»'. + (l-|j!? ('^»)-

Substituting for x^ its value from equation (727), and for /,

its value -^, and reducing,

-M==W^ mn. yi + ^ cos. 7t+ ^1-^--^^--)^.
Integrating between the limits 0 and i, and observing tbat
when (=0, -77=0 ; the time being supposed to commence
with the motion of tlie prism PQ,

• Art. 9fl. Equations (13) and (74).
t Church's Int. Cal. Art. 133.

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568 IMFAOT.

Integrating a second time between tlie same limits,

Ixow when tlie motion of tlie second prism ceases -57=0 ;
whence, if the coiTesponding value of t "be represented by T,

A(l-cos. 7T)+BBin. r T + ( 1 - .y^-)T,X7T=0.(731).

To detemiine the constants A and B, let it be obseryed
that the motion of the priem QP cannot commence until the
pressare Q of the impinging prism upon it, added to its own
weight W,, is equal to the resistance P opposed to its motion.
So that if e be taken to represent the value of a;, {*. e. the
aggregate compression of the two prisms) at that instant,
then, substituting for Q its value from equation (725), - +
■W,^P ;

,.o = (P-W.)>. = (P^W,)(jl;^+5ij-;....(W2).

Now since tlie time t is supposed to commence at the
instant when this compresaion is attained, and the prism PQ
is upon the point of moving, substituting the above value of
e for jc, in equation (727), and observing that when ai=c,

i=0, we ohtain (F— "W"-)X=B+ ,.°L ; whence by substitu

tion from equation (728), and reduction,

(P-W,-W,> / P >

So long as the extremity P, of the prism impinged upon,
is at rest, the whole motion of the point B arises from the

compression of the two prisms, and is represented hy -^

./Google

(ea[iiatioii 'TSl). Differentiating, therefore, equation (727),

asauming (=0, and sulastituting v foi'-jf' we obtain v=yA',

whence it appears that the value of A is detei-nained by
dividing Uie square root of the second number of equation
(731) by r

Substituting for A and B their values in equations (731-3)

?(l-oo=.rT)+>.W.(^Z^-l)sm.rT +

Eeducing, and dividing by the common factor of the two
last terms,

■i)(l— cos.yT)

+ sin.7T-7T=0 (734.)

Snbstitutirig for A and B their values in equation (730), and
representing by D the value of !»,, when i=T,

.... (735).

The value of T determined by equation (734) being sab-
gtitnted in equation (7S5), an expression is obtained for the
whole space through which the second prism is driven by
the impact of the first.*

* The method of the dbove inTeaUgation ia, from equation (726), nearly the
Bame with that given bj Dr. WheweU, in the last edition of hia Mechaniea; the
pcinoiple of the iQTeatigalJon appears to be due to Mr. Airej. If the tkIub
of 7, SB determined by equation (T28), wera not exceedinely great, then, since
the value of Tiain all practical cases eiceedingijsniall, the value of j'T would
in all ci^ea be exceedingly amatl, ejid we might approximate to the T^ue of
T in equation (186), bj substituting for cos. yT and sin. jT, the twc flPHt temM
of the eipanBiona ot those fimctiona, in terms of yT.

, Google

, Google

EDITORIAL APPENDIX.

Note (a).

. ! direction defined (Art. 1), we have also to take

into consideration, in estimating the effects of a force, its
pOfMt of ajpplication, or the point of the hody where it acts,
either directly or through the medium of some other body,
aa a rigid bar, or an inextensible cord in its line of direction ;
the point on its line of direction towards which the point of
appRcation has a tendency to moTC ; and finally the inten-
stt)/, or magnitude of the force as expressed in terms of some
settled unit of measure.

Note (i1.

This result of experiment also admits of the following
proof: Let A be the point of apph-
cation of a force P, and let tliis point
be inyariably connected with another
point B, in the line of direction towards which A tends to
move from the action of P ; suppose now two other forces
P, and P,, each equal to P, to be applied ; the one at A, in
a direction opposite to P, and the other at B, in the same
direction as P ; the introduction of these two equal forces,
acting m opposite directions, will evidentlyin no wise change
the direction -or intensity of P ; but as P, is equal and oppo-
site to P its effect will be to balance the action of P at A,
whilst it leaves P^ to exert an action at B precisely t!ie same
as P was exerting at A before the introduction of P, and P,.

Note (e).
two forces P, and P^, applied to the same point A,

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573 editor:

tlie dii'ection of the one being AB, that of the
other AC; no was these forces make an angle
with each other, it ia evident, as the point of
application can move hut in one direction, and
as it is sohcited to move towards B and 0 at
the same time, that it must move in some
direction which is coincident with neither of
these; this direction, it is equally evident,
1 the same plane as the directions AB and AO, for
J argument in favor of a du'ection assmned exterior
e and on one side of it which would not eqiially
apply to a symmetrical direction asaunied on the other side;
it IS also evident that this direction must be some one AF
within the angle formed by AB and AO, for the point, if
solicited by P, alone, would take the direction AB, and as it
cannot take a du-ection to the left of BD, as there is no force
that solicits it on that side, and, for like reasons, cannot take
one to the right of OE, it must therefore take tlie one
assigned somewhere within the angle BAG.

Now suppose further that P, and P^ are equal, it is evi-
dent that the direction assigned to their resultant, or that of
the motion of their point of application, must he the one
which bisects the angle BAG, for the argument for any
direction on the left of this line would be equally cogent for
the like position on the other side.

If P, and P, are uneq^ual then will the direction of their
resultant divide the angle BAG unequally, the
smaller portion being next to the greater force ;
for suppose P, divided into two portions, one
of which P shall be equal to P, ; P and P, can
be replaced by their resultant E,, the direction
of whidi AF bisects the angle BAG ; we shall
then have two forces Kj and the remaining
portion of P„ Uie resultant of which R must lie
somewhere witldn the angle BAF, and there-
fore nearer to Pj than to P, ; but R is the resultant of the
two forces P, and P,. Therefore, &c.

Hence it is seen that two forces whose directions foi-m an
angle between them and meet, 1st, have a resultant ; 2nd, that
the direction of this resultant lies in the plane of the two
forces ; 3d, that it passes through the point where the direc-
tions meet, and lies within tne angle contained between
them ; ■ith, that it bisects this angle when the forces are
equal ; 5th, that when the forces are uneq^ual it divides this
angle uneciually, the smaller portion being next to the greater
force.

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KDITOKIAL APPEISDIX. 573

Wow as the two forces P, and P, can be replaced by their
resultant E, and aa the effect of this will be the
JK same if applied at any point F in its line of

/I X^ direction as at the point of application of the
/ L ' two forces, it is evident, if we ti'ansfer P, and P,
/ /r\ ^^*' *° ^' pi'sserving thftir new parallel to theii
^* / ^» original directions, that they, in torn, can be made
/ tg^ to replace It. It thus appears that the point erf

J apphcation of two forces may be transferred to

' Miy point of the line of direction of their result-

ant withoiit changing the effects of th^e forces, pi-ovided
their new directions are kept para,llel to their original ones.
It is npon the preceding propositions, in themselves self-
evident, that the mode of demonstration, known as Duchay-
la's, of the proposition, termed the parallelogram of forces,

Note {d).

AVlien two parallel forces are applied to two points inva-
riably coimected, their resultant can be foiuid by applying
J-he propositions in (Arte, 1, 2, 3).

Let F, and P, be two parallel forces applied at the points

A and B invariably connected, as by a

oc^S-yUM rigid bar. Let two equal forces Qj and

f/f^^ Q, be so applied, the one at A the other

rZ-fm \ at B, as to act in opposite directions

j^Qs along AB, These two wiU evidently

have no effect to change the action of
^HJ / P, and P,. Now the two forces P, and

'4,-^ Q, applied at A will have a resultant Kj,

the intensity and direction of which can
be found by the preceding method. In Bke manner the
resultant K^ of P, and Q, can be obtained. Now the forces
being replaced by their resultants, the equilibrium will still
subsist, and the effect will remain the same whether R, and
B, act at A and B, or at o their point of meeting. But aa
R, and E, can each be replaced by tlieir components at any
point of their direction, let these components be transferred
to the point o. In this position Qj and Q, will destroy eaoJi
other, whilst Pi and Pj will act in the same direction along
oC and parallel to their oiiginal ones, with an intensity equal
to their sum P,-l-P,.

Now from the similar triangles AoC, Tom; and BoO, 80»,
there obtains,

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57* EDirOEIAL

om:mr::oG:GA,orJ',:Q,::oC: CA.
ns : on :: CB : oC, or Q, : V, :: CB ■. oC.

Multiplying tlie two last proportions, there obtains,

r, : P, : : CB : CA,
and

P, : P, : P,+ P, :: CB : OA : CB+CA or AB.

Prom this we see that two parallel forces acting in tha
same direction, 1st, have a resultant which is eqnafto their
STun ; 2nd, that the dkection of this resultant is parallel to
that of the forces ; 3d, that it divides the line joining the
points of application of the two forces into parts reciprocally
proportional to the forces ; 4th, that either force is to the
resultant as the portion of ^e line between the resultant and
the other force is to the total distance between the points of
application ; 5th, that the foregoing propositione hold true
for any position of the line AB with respect to the two
parallel forces and their resnltant.

When the two forces act in opposite directions at the
poiuts A and B, by following

^^if"^ ^ ^^^ process, we obtain the

'Tf**'/]-^ ..-'-""'^ ^'^ *wo resultants B, and E„
/ LC I .^"^ ' which being prolonged to

k^^^v i}a&\T point of meeting o we

'W~.:f^l. fis^ again replace them by

; /; their components Pj, Q„ and

j/.J P„ Q,; of which P, and P„

™ acting parallel to tlieir ori-

ginal positions but in opposite directions, will have for their
resultant P,— P,.

Now pi-olonging the direction of this resultant until it
meets AB prolonged at 0, there obtains as in the preceding
case, from the similar triangles AoC, rom. and BoC, son,

om : mr :: oC : OA, or P, : Q, :: oC : OA,
ns:m::GB: oG, or Q, : P, :: OB : oG,
hence,

P, : P, : P -P, :: OB : OA : OE-OA or AB.

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EDITOEIAL APPENDIX.

5T5

t may be assumed, ae eelf-tvident,
that any resultant can be replaced
by two eqmvalent components,
■without di&tarbing the equilibrinm,
and that each of these in turn
may be replaced by two otJier
ecLuivalent components, and so on
for any number of components ; etill lite compositions and
resolutions of forces sre of such frequent occurrence in esti-
mating the pressuree, or strains on the various points of any
mechanical contrivance, as a machine, a frame of timber,
&c., arising from a resultant prtssure, that the student can-
not be too familiar with the processes of effecting such com-
positions and resolutions.

To show by a simple illustration this truth, let the result-
ant AE be replaced by its two equivalent components AP,
and AP, in any assumed positions ; and let each of these
components be resolved into two others, AQ„ AB, for AP, ;
and AQj, AR, for AP„ taken respectively pei-pendicular and
parallel to AE. Now it is evident, from the figure, that the
two components AQj, AQ, of this last resolution are equal
and opposite in direction, and therefore destroy each other ;
wliilst the two AK„ AR, act in the direction of AR, and
their simi is equal to AR. The same may in like manner be
shown for any number of sets of components by which AR
might be replaced.

Note («).

If the point o from which perpendiculars are drawn to the
directions of two forces P, and P„ is
taken on the direction of their re-
sultant, then will jn.'P,=m'Pj,

For from o draw the perpendicu-
lars om, on, to P, and P,, and join
the points m and n of their inter-
section. The quadrilateral Am<m,
hainng the angles at -m and n right
angles, can be inscribed in a circle, therefore the two an^es
at m, and A, subtended by the chord on, will be equal. In
the triangles rnon and ABO, the angle o is the supplement
of the angle A of the quadrilateral, and B, being the adja-
cent angle of the parallelogram, is also the supplement of A;

./Google

the two triangles, having two angles equal, are eimilar,
therefore,

AE : 130 :'. om : on, or P, : P, :: om '. on;
hence

P, X om=V^ X on. Therefore, &c.

From this proposition the relations of two parallel forces
to their resultant can be readily deduced from the limiting
case of the angle mon of the triangle ; for from the two simi-
lar triangles there obtains as before

P, : P, : E or h.Q :: om : on : mn.

Kow as tMa is true for any value of the angle o, when it
becomes 180°, tlie forces Pj, P^ having the same direction,
and their resultant E become parallel; the perpendiculars
om and on become portions of the line «wi ; and, as 7iin=-om
+on, it follows, from the above proportion, that E=;P,+Pj.

Wlien P, and P, have opposite directions, we can suppose
the force Pi, for example, and its perpendicular turned about
the point o in the plane of the forces until the point m falls
on the prolongation of on on the opposite side fi'om o, in
which position P, and P, again become parallel, but act in
opposite directions. During this rotation of P„ the resultant
still passes through o, and there still obtains

~P, ■ 7, :: 'R : om : on : mn ;

but, as mm now is equal to om—on, it follows, from the
proportion, that R=P5— Pj, Hence the same relations
between P„ P^ and E as already established, Note {d).

KOTE {/).

Otherwise, since in any number of forces in equilibrium
either of them is equal and directly opposed to the resultant
of all the rest, the whole may be replaced by these two
■withont disturbing the equilibrium. If now through the
point of apphcation of these two we draw any two lines at
right angles to each other, and then resolve each of the two
forces into two components parallel to these two lines, it
will be at once seen, from the diagram, that the like com-

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EDITORIAI. Al'PiaillS. 571

jtonente will be equal and opposite to eacli other, and this
would evidently 1)6 the same for the components of the ori-
ginal forces resolved in tlie same manner, otherwise tliere
would be a resultant for all the forces, which is contrary to
the supposition of an eq^nilibrium.

Bemark, — As this method of resolving a system of forces
into sets of components pai-allel to any assumed rectangular
axes, in order to determine their algebraical values, is of
frequent use, in simplifying the numerical calculations
necessary in the applications of the principles of statics, the
student should mjuce himself perfectly familiar with the pro-
positions that precede and follow Art. 11.

Note ig).

Otherwise, join DE which will be parallel to AC, thns
forming with it and the lines AD and CE two equi-angulM"
triangles, from which there obtains

DE:DG:: AC: AG;

but DE=iAC, therefore DG=;iAG=iDA.

KOTK(A).

Otherwise, join Gil which, as AG and CH intersect, will
be in tlie same plane with them and with the line AC. As
AH and C6 are respectively \ of the lines di'awn from A
and C to the middle of BD, it follows that GH is parallel to
AC and forms wiiJi it and the lines AG and CH, by their
intereeetion at K, the two eqtu-angular triangles GKH and
and AKC, from which there obtains

GH: GK :: AC: AK,

but GH=iAC, therefore 6113=1 AK=|AG.

Note (»).

As the methods employed in (Art. 45, &c.) to represent,,
by geometrical diagrams, what are termed the laws of
motion, or the relations which exist at any two given
3T

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578 EDITORIAL ATPENDIX.

instants "between tlie velocity, the space, and the time of a
body's motion, altlio\igh veiy simple in tliemeelves, are
Bometimee found to offer difficulties to the student, particu-
larly 83 to the representation of spaces by areas, a few addi-
tional mai'ks on tliis point may not be here misplaced.
Taking, in the iirst pliuie, tne case of a body M. moving in
J, a rectilinear path from Atowards

'' '~ \ B with a uniform motion. Ac-

cording to the definition, the
I body mil move over the suc-

H/^ ' ' rf ' ~ ceesive ecLual portions Ab, ic, cd,

^ &c., of its path in equal succes-

sive portions of time, however small or great these portions
may be. Taking now any portion of time as a unit, as a s&cond,
a rwmide, &c., and aupposing Ah the portion of its path, or
the space through which the body has moved dunng this
unit. Ah will represent what is teimed the velocity, or rate
of motion of the body ; and when the path itself is expressed
in terms of any linear unit, as a foot, a yai'd, a mile, ifcc, the
number of these units in Kb will measure the velocity ; for
example, if the unit of path, or space is a foot, and there
were four of these units in Ah, and tiie unit of time is a
second, then the velocih' woidd be tenned a velocity of four
feet per second, &c. Supposing the body to stai't from A,
with tiiis velocity, it will successively move over distances,
each of four feet in length, along its path, in successive
■seconds of time ; consequentiy any distance, or apace, as Ad,
■will be equal to Ah taken as many times as the number of
seconds elapsed from the time the body started from A until
.'it reached d\ or, in other words, tiie number of units in the
.■6|pace Ad is expressed by the abstract munber obtained by
^multiplying the number of units in the velocity by the num-
sber of units in tiie time. This, like all other similar pro-
'duKta, can be espre^ed algebraically, or geometrically ; but
bywbatever symbol expressed, the signiffcation is the same.
For .example, on any two lines, as AB and AC, taken at
right angles, set off any number of equal pai'ts as A5, he,
cd, &c., as units of time, and on AC any number also of
equal parts, which inay be the same in length, or otherwise,
as those on AB, to represent the units in which the velocity
is expressed. Suppose the latter to be composed of the four
unite Am, mn, &c. ; and that the number of units of time
censidered is three ; on the lines Ad, AC construct the
Teetangle AD; then is tiie area of the rectangle said to
i the .space corresponding to the velocity and tiuie

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579

here assmned ; that is, the number of imits in ai-ea of this
rectangle, expressed in terras of tJie luiit of area on Kb aud
Am for example, is equal to the number of units of space.
In like manner the area of the rectangle AE expresses the
space corresponding to the velocity and the time Ae, &c.
In nniforraly varied motion, aa the velocity increases in
the same proportion, as the time
increases, or, in other words, aa
the augmentations of the velocitj
for equal intervals of time ia the
same, these relations between the

i^T 1,1 I J, times, velocities and spaces, can,

^ ** ^ '" ui lUre manner, be expressed by

a geometrical diagi'am as follows : On any line, as AB, set
off a number of equal parts as AJ, 6c, ed, &c., to represent
equal inteiwals of time ; at the points 6, c, d, &c., having
drawn perpendiculars to AB, set off on them distances Jm,
en, do, &c., to represent the con'esponding velocities; in
which cn^2hm do=dhm ; or Ad : A.C : Kb :: do '. en, '. Sm,
&c. Now, as the same relations obtain between all the dis-
tances set off on AB and their corresponding perpendicu-
lars, it follows that tlie line AC, drawn throxigh the points
m, n, 0, &c., is a right line, and that the triangles Ahm, Aora,
&c., are thei'efore similar. As the relations between the
times and velocities are true, however great, or however
small the equal portions of time may be assumed, let us sup-
pose these portions, as AS, bo, cd, to be taken so small that
the velocity of tlie body during any one of them may be
considered uniform, and as a mean between what it actually
is at the commencement and end of this portion ; that is en
and do, for example, representing tlie actual velocities at the
beginning and end of the interv^ of time represented by tid,
then ^ (cn + do) represents the mean, or unifonn velocity
during this intervifl. This being premised, the number of
units of space over which the body will pass whilst moving
with a mirform velocity, expressedby i (en+do), during the
interval od, wdl be represented, according to tlie preceding
proposition, by cdx-k {en+do), but this also expresses the
area of the trapezoid odno ; and as the same is true for all
the like trapezoids it will also be true for their sums, or for
the triangles, aa Ado and Afq for example, tlie areas of which
are equal to the sum of the areas of the trapezoids of which
they are composed. Supposing the body to move from a
state of rest with a unifoi-mly accelerated motion, and that
at the intervals of time, represented by Ad and Af, its

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580

i-espectiFe Yelocities are tfo and^, tlien will the number ot
units of space wbich tlie body will have moved over in these
two intei-vals be respectively expressed by the mimber of

two uitei-vals be respectively expressed by t
units of area in the triangles Ado and A/^. As the trian-
gles are similar their areas are as the squares of their like
sides ; it therefore follows tliat in unifonnly vaiied motion,
the spaces are as the squares of the times, or as the sqnai'ea
of the velocities.

As do represents the velocity acquired dining the time
Ad, supposing the body to have moved from a state of rest,
and the number of units of area in the triangle, Ac^o repre-
sents tlie corresponding number of units of space, it follows,
tliat if the body had moved, during the same interval, with
the velocity do which it actually acquired in it, the number
of units of space it would then have pa^ed over would have
been represented by the number of units of area in the rect-
angle Ao, constracted on Ad and do. But, as the area of
the rectangle is double that of the triangle, the space that
would have been passed over in the euppceed case would
have been double that passed over in the actual case.

If we take any poiiioii, as Ae, to represent the unit of
time, then the cori'esponding perpendiculai' ep will represent
the velocity, or the quantity /used in (Arts. 46. i1) fol-
lowing.

Note (J).

As the propositions under this head, and those under the
heads of Accumulation of Work in a Moving Body (Art. 6i)
and Principle of Vis Viva (Art. 129) constitute the basis of
what may be tenned Industrial Mechanics, or the applica-
tions of the piinciples' of abstract mechanics to the calcula-
tion of the ^ects of motive power transmitted by machines
and employed to prodnce some useful mechanical end, it is
very important that the student should have a clear and
dennite apprehension of their signification in this point of
view. . Work, as here deiined, supposes two conditions aa
essential to its production : a continued resistance, or obstacle
removed by the action of a force, and a motion of the point
of application of the force in a direction opposite to that in
which the resistance acts. Its measure is expressed by the
product arising from iuuitiplying the number of units of the
resistance, or of its equivalent torce directly opposed to it,
by the number of units of path which the point of applica-
tion of this force has described during the interval consi-

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EDITORIAL APPENDiX. C81

dered, in wliicii the force acts to overcome the reaistaneti.
It follows tliat the work wiU be 0 when this product is 0 ;
that is, when either of the factoi-s, the resistance, or ite eqai-
valent force, or the path described, is 0.

In eetimating work, that which is external and which alone
generally we iiave the means of measuring, is alone consi-
dered. For example, if with a flexible bar a person attempts
t« push before him any obstacle, the first eflect observed will
he a certain deflection, of the bar, during which the Iiand, at
one end of the bar, will have moved foi-ward a certain dis-
tance in the direction of the point of apphcation at the other,
producing an amount of work which is expressed by the
product of the preesm-e, or force exerted by the hand^ sup-
posing this pressure to remain constant during this period,
and me path described, in the line of direction of this pres-
sure, by the point where it is apphed. Dm-ing this period,
as the obstade to be moved has remained at rest, no path
has been described by the point where the bar rests against
it, therefore, according to onr definition, no work has been
done upon the resistance. The effect produced by the
pressure has been simply to bend the bar, and the wort ia
therefore due only to the resistance ofi"ered by the molecular
forces of the material composing the bar to the force that
tends to bend it. This portion ot the work, although in this
case we have the means of measuring it, being what may be
termed internal, is not taken into the account in estimating
that duo to the resistance to he overcome, which woidd have
been the same had a perfectly rigid bai- been used instead of
the flexible one.

In like manner, when an animal caiTies a burthen on his
back from one point to another on a horizontal plane no
work is produced according to our deflnition ; for no resist-
ance has been overcome in the direction in which the bur-
tlien has been carried, and therefore the product that repre-
sents the work is 0. The work in this case, as in that of the
flexible bar, is internal ; and similar to that arising from a
bm'then borne by an animal wJiilst standing still ; and there-
fore although both of them may be very useful operations
and have a marketable value, still they can neither be mea-
sured by the standai-d by which it is agreed to estimate
work.

Every mechanical operation perfoi'med by machinery pre-
sents a case of work. Take for example the simple operar
tion of i)laning, in which the hand moves a plane, which is
but a rigid bar to which is fixed an iron tool like a chisel for

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583 EDITOIIIAL APFEKDIX.

removing enceeesiTe thin portions from the edge, or ani'faee
of a board. In this case the resistance offered, and which is
seneihly in the same direction as the power applied, is that
arising from tlie cohesion of the fibres of the material, and
is measured by the pressure applied ; the path which the
point of application of the ii'on tool describes is the same as
that described by the hand ; and the work will be expressed
by the product of tliese two elements, eacli estimated in
terms of its own unit of measure. The case of the common
grindstone presents an example of a rather more compUcated
character. Here the instrument to be gi'ound is pressed
against the periphery of tlie stone with sufficient tbrce to
cause a certain resistance to any power however applied to
put the stone in motion. The direction however in which
this resistance acts at the point of application is in the
dii'ection of the tangent to the periphery at this point, and,
in one revolution ot the stone, it will describe a path equal
in length to the circle described by the point of application.
The work tlierefore for each revolution will be the product
of the resistance, estimated ia the direction of the tangent,
and the circumference described by the point of application.
It mav be as well to remark, in this place, that although
the work done in overcoming the molecular resistances of
the materials by means of which the action of a force or
pressure is transmitted, as in the example above cited of a
flexible bar, is not taien into accmmt in estimating tlio
extei-nal work, there are cases in which tliis work constitute
the entire wort done, and which again is reproduced in
external work ; as for example in the cases of the common
bow used for projecting arrows, and the springs by which
the machineiy. of some time-pieces is moved. In each of
these the resistance offered by the molecular forces of the
material is overcome by the action of some extenial force,
whose point of application is made to describe a given path ;
by this action a certain amount of work is expended in
bringing the spring to a certain degree of tension which,
when the force is withdrawn, will reproduce the same amount
of external work in an opposite direction to that in which the
force acted.

Note (A),

The work of a pressure of constant intensity acting in the
same direction as the path described by its point of applica-
tion may be represented by a geometrical diagi'am m tlie

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5S3

Eiine way as that used for representing the space described
by a body moving with a uniform velocity in any giver
time ; by eonstnietmg a rectangle, one side of whicli repre-
aenta the mimber of units of force, the other the number of
units of path ; the number of units of area of the rectangle
will express the number of unite of work.

KoTE {I).

The method given (Art. 51) for estimating, by a geometri-
cal diagram, the work of a pressure which varies in inten-
sity at different points of the path described in its line of
direction by its point of application, finds its application and
has to be used whenever the'"e is no geometrical law of con-
tinuity by which the pressure can be expressed in terms of
the path ; and, even when such a law obtauis, it is some-
times found to be a more convenient method of obtaining an
approximate value of tlie amount of work than the moro
rigorous one expressed by the formula

n=fp.

in which TJ can be rigorously found wheiiever P, which being
a iunction of S can be expressed algebraically in terms
ot it

Ab an example of these two methods of estimating tlie
woik of a variable pressure, aeting^ in tlie
direction of the rectihnear path described by
itt> point of application, let the familiar case
of the action of steam on the piston of tiie
hteam-^ngine he taken.

Let ABCD represent the steam-tight cy-
linder in which the piston is driven from the
pobition at a, at one end, to c at the other, in
the direction of the axis ac, of the cylinder,
by means of the pressure of the steam on the
end of the piston. Let iis suppose that the
■^ -" steam acts with a constant pressure, repre-

sented by Pi, whilst the piston is driven through the portion
ha of the path, and, having reached tliis point, the commu-
nication between the cylinder and the boiler being then cut

./Google

off, that the steam already admitted acts, through the re-
mainder of tlie path described by the piston, by what is
termed its expansive force, in which the preesm'e continually
decreases, as the piston approaches the point c. Let ne 8\ip-
pose that the law of variation of this pressure on the pisto]]
at different points is such that the pressm'es at any two
points are invereely proportional to their distances from the
point a. P, then denoting the preaeure wlien the piston is
at 5, let P denote the pressure when it has reached another
point 0 at a distance S from a, and 8, and . S, denote the
lengths as and dh, then according to the above law there
obtains

P, : P ;: S : S„ therefore P=P,^.

Let tlie elementary portion of the path be denoted by (?S,
then by multiplying the variable force '^'^ the elementary
path there obtains

which may bo termed the elementary worlc, or in other
words, the work done whilst the variable pressure acts
through the elementary path, during which period the vari-
able preesui'e may be regarded as constant.

To obtain the total work whilst the variable pressure acts,
from 5 to 0, or through tliG patli Sj— S„ there obtains

TJ =f'^d^ = P,S,y ^=P,S,(log.e S,-log., S,).

If instead of the exact work due to the expansive force of
the steam, and which is given by the foregoing fonnula, an
approximate value only was required, it could be obtained by
a geometrical diagram as follows.
"" Having set off to any scale a num-

ber of units representing the path
"■■•-.^ Jc, calculate the pressures at the

"T-- points 5, 0, and at the middle point

"f 0, for a first approximation, Hiat
I 1^ at h will be simply Pjj that at o,

P,|i, and that at o, P,- ^'

■ i(S, - SO

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APPENDIX. 58S

Having drawn perpendiculars to &c at h, o, and c, set off on
tliein the distances bm, on and cy respectlvelj equal to the
corresponding pressures, estimated in terms of the unit of
pressure, and according to any given scale. Join mn and
up ; the number of units of area, in. the tigiire tlius formed,
estimated in imits of path and pressure, will be an approxi-
mate value of the required nnniber of units of work.

The greater the number of parts hito -wMcli he is divided and
the corresponding pressures calculated, the nearer will the
enclosed area approach to the ti'iie value of the work.

The mean pressure, or that force which, acting with a
constant intensity along the same path as tliat described by
the point of appUcation of the vaiiablo pressure, would give
the same work, is found either by dividing the result of the
integration by S,— Sj, or by dividing the ai-ea in the last
method by 5c..

IlJ'oTE (m).

As an example of the manner of obtahiing the work done

_,4,.__ by a constant pressure acting always

/■^ j ''••., in parallel directions whilst its inclina-

/ p. Ac tion to the path d^cribed by its point

/ oif-^^'^V '^^ application is continually vaiying,

\ \m!->^ jgj; (J,Q y/i^ known mechanism of the

y \ crankai'mandcomieetmgrodbetaken.

"~--, ..••' '■-■ ■ Let O be the centre ai'ound which the

^ crank arm is made to revolve, by the

application of a constant pressm'e P„
* transmitted through a connecting rod

CD, all of whose positions duinng the motion are pai-aliel to
the diameter AB. The path described by the point of ap-
plication 0 will be the circumference of which OC is the
radius, and the inclination of P, to this path will be the
variable angle DON, between its direction and the tangent
to the circle at C, of which the variable angle AOC, tliat
meiisnres the inclination of the crank ann to the diameter
AB, is the complement. Denote this last angle by a, and
the length of the crank ai'm 00 by 5. Now decomposing
P, into coniponents in the direction of the tangent OF and
the radius 00, we obtain for the first P, sin, a, and for tlie
second P^ cos. o, of which P, sin, a is alone effective to pro-
duce work, since P, cos, a acts constantly towards the fixed
point O without describing any path in the direction of its

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030 EDITOEIAL

i:utioii. Eut lihe elementai-y path deecnbed Ly tho poinl
Of application is evidently l>da, the infinitely email ai-c Ore of
ilio circle. The elementary work of the variable component
y, sin, a will therefore be expressed by

P, sin. a X l)da.

The total work for any poi-tion of the path, as AC, will
llierefore be

f^

P,sin. (i Wo:=Pi5(l— COS. a)=P,^ver, sill. a.

and for a=7:, it becomes

P,x2&, orP^xAB;

a resnlt wliich might have been for^een, since AB is the
path described by the point of application of Pj in its line
of direction, whilst the actual path is the senii-circxmife-
rence AOB.

As Qn=.'bdA^ if through n a perpendicular nirb is drawn to
OD, the line of direction of P„ the distance C*«. is evidently
the projection of the elementary path actually described on
the line of direction of P„ and is therefore the corresponding
elementary path of P, in its line of direction ; but Cm^Cn
sin. a:=hda sin, a. Denoting AB by h, then Cm^dh ; and
there obtains

dh=Ma sin. a ; and P, dk=F, 7>da shi. o;

and

y \\dh^'2Ji = fv,lzuv.ada=y^x'2,h.

A result the same as is shown to obtain by the preceding
proposition.

To find the mean, or constant pressure which, acting in
the direction of the circular patJi, would produce the same
amount of work as the vaiiable force does in acting through
ttie Bemi-cireumference ; call Q this mean force, ite path
being wS, its work will be Q X w5 ; and as this is to be eq^oal
\a the work of P, sin. a, there obtains

Q X -nh-V, X 25, hence Q=P, - =0-6366 P, nearly,
for the value of the force.

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EDITOEIAL APPKNCIX. 58T

It may "be well to observe here that the mean pi-e^ures
have no farther relations to the actiial pressures than as
numerical resnlts which are frequently used instead of the
actual pressures to facihtate calculations ; and also as a
means of comparing results, or work actually obtained from
a force of variable intensity, at diiFerent epochs of its action,
■with what would have been yielded at the same epochs by
the equivalent meaii force.

To show the manner of making the comparison in thie
case, let us take the two expressions for tlie quantity of work
due the mean force, and also to the vmiable component, for
a portion of tlie path corresponding to any angle a. Since

Q=P,-, its work coiTesponding to a will be

Tlie corresponding woi'k of the vaiiable component V^ sin. a
will "be

P,J{l-cos. a).

The difference therefore between these two amounts of work
will be

VJ>^-'Pfi(l-eos.a)=V,h j-^-l + cos. aj

Now this difference will be 0 for the following values of a,

a=0, a= ^, and a=7T.

The maximum value of this difference can be found by the
usual method of differentiation and placing the firat differ-
ential coefficient equal to 0. Performing this operation,
there obtains

sin. a=-=0-6-366;

the corresponding values of a, being respectively

a =1 0-21964 7r,and o. = Tr-0-219647r.

Substitaiting these values of a and the corresponding values
of cos. a in the preceding expression for the difference there
obtains, for the iirst,

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EDrrOKlAL APPENDIX.

PJ> f^;^ - 1 + eos. a) = l',S (2 X 0-219G4-1 + |/l _ i\ =

+ 0-21039 P,&;
End for the second,

P /, p® -1 + COS. a j =P,?.(3-3 X 0-21964 - l-|/l - ^) =

~0-31039rA

Prom tiese two expressions it is seen that the gj-eatest excess
of the work of the mean force over that of the other would
be + 0-21039 P,5= + 0-1052 xP,25, or about j\ of the totd
work of P, coiTesponding to the path 2& ; wliilet that of the
workofP, over the meanforce, represented by — 0-21039P,5,
is the same in amount,

K now we suppose the direction of the constant force P;
to be changed, when its point of application reaches tlie point
B, 80 as to act parallel to tlie direction BA until tlie point of
appHeation ari-ives at A, it is clear that the work of P, due
to the path described from B to A will also be expreseed by
P, X 25, so that the work due to an entire revolution of the
point of application will be P, x ih. As the mean force will
evidently be the same for the entire revolution of the point
of application, it follows that the gi-eatest pc^itive, or negar
five excess, as stated above, will be 0-0536 xP,4&, or /^ of
the work for one entire revolution.

It is thus seen that although the work of the effective
Fariable component P, sin. ffl of P, is not, like that of the mean
force, unifoiTQ for equal paths, still it at no time falls short
of nor exceeds the work of the meap force by more than
about 5V of *^^® entire work for each revolution. "Were any
mechanism, as that for pnmping water for example, so
arranged tliat either the constant force P„ or a mean force
equal to 0-6866 P„ acting as above described, were applied
to it, the quantity of water delivered by the one would at no
time exceed, in any one revolution, tnat delivered by the
other by more than ^V of tlie total quantity delivered by
either during the entire revolution.

KoTE (n).

If Pj, for example, were tlie resultant of tlie other pres-
em-es, its compouent 1'^ cos. a, would be equal to the alge-

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ECITOEIAL APFESDIX. 5S8

braiG sum of the components P, cos. a„ P, cos. a,-, &c., of the
other pressures P„ P,, &c. ; the work therefore of P„ euti-
mated in tiie direction of the given path AE, and corres-
ponding to any portion of this patli, will he eqnal to the
algebraic sum of the work of the other pressures ?„ P„ &c.,
corresponding to the same portion of ttie given path.

"NoTJl (<?).

Since at the point E, taken as the point of application, the
line of direction of the pressure becomes a tangent to the
are described with the radius OE, it follows tliat the infi-
nitely small arc desciibed with the radius OE may be taken
for the infinitely small path described by the point of appli-
cation in the direction of the tangent. Denoting by da the
infinitely small angle described by the radius OE, thon
OE X da will express the infinitely small patli, or arc ; and
P X OEda will represent the elementaiy work of the
pressure.

If the pressure remains constant in intensity and direction
duiing an entire revolution of the body about 0, then will
tlie work of P for this revolution be represented by
P X circum. OE.

The tei-m living force is moj-e generally used with us by
writers on mechanics instead of its Latin equivalent vis viva,
to designate tlie numerical result arising from multiplying
the quantity denominated the mass of a body by the square
of the velocity with which the body is moving at any
instant. It will be readily seen that this product does not
represent a pressure, or force, but the numerical equivalent
of^the product of a certain number of units of pressure and a
certain number of units of path. The one magnitude being
of as totally a distinct order from the other as an ai'ea is
different from a line, and therefore having no common unit
of meaam-e.

Besides this expression, which sei-ves no other really use-
ful pui-posesthan as a name to designate a certain numerical
magnitude which is of constant occun^euce in tlie subject of
mechanics, tliere is another also of frequent use, termed

./Google

EDrrOEIAL APPENDIX.

qxitrntity of motion, wiiieh ia tlie product of tlie ina=8 and

the Telocity, or — v. This ie also termed the dynamical

measure of a foroe in contradistinction to pi-eseure, as usually
estimated, which is termed the statical measiire of a force.

Note (q).

In estimating the acenniulated work in the pieces of a
niachine whieli have either a continuous or a reciprocating
motion, of rotation it ia necessary to find expressions for the
moments of inertia of these pieces with reepect to their axis
of rotation, and this may, in aU eases, he done, within a cer-
tain degree of approximation to the tnie value, hy calculat-
ing separately the moment of inei-tia of each of tlie compo-
nent parts of each piece and taking their sum for its total
moment of inertia, on tlie principle tliat these moments may
he added to or stibtracted from each other in a manner
similar to that in which volumes, of areas are found from
their component paita.

In making these approximate calculations, whicli in mauy
cases are intricate and tedious, it will be well to keep in view
the two or three leading points following, with the examples
given in illustration of some of the more usual foi-ms of
rotatingpieces.

1st. The general form for the moment of inertia of a hody
rotating ai'ound au axis parallel to the one passing through
its centre of gravity as given in equation 58, (Art. 79) is

I,=7.'M + I.

Now if the distances of the extreme elements of the body
from the axis passing through its centre of gravity are small
compared with that of h, tlie distance between the two axes,
the second tenn I of the second member of this equation
may be neglected widi respect to the first, and A'M be taken
as the approximate value of the required moment. This
consideration will find its application in manv of the cases
refen'ed to, as, for example, a\ tliat of finding tke moment of
inertia of liie portion of a solid, like the exterior flanch of the
beam of a steam-engine, the volmne of which may be approx-
imately obtained by the method of Ovldirvus (Art. 39.). In
this case, A representing the area of the cross section of the

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EDITOErAL APPIKDIX. 591

fianch, and s the path wliich its centve of gravity would
describe in moving pai'allel to itself in tlie direction of tbo
flanch around the beam, any elementaiy volume of the
flanch between two parallel planea of section will be ex-
pressed by Ads. Now tlie moment of inertia of this elemen-
tary volume from equation 58 is

ijr^ds+l ;

I, =A

in which the first term of the eecoud member, which
expresses the sum of the elementary volumes Ads into the
squares of their respective distances r from the axis of rota-
tion, may be taken as the approximate value required ; inaa-
mueh as I, the sum of their moments of inertia with respect
to the parallel axes through their centimes ofgravity, may he
neglected with respect to the first term. The problem will
therefore reduce to finding tbe moment of inertia of ^e line
represented by s, which would he described by the centre of
gravity of A, with respect to the assumed axis of rotation,
and then multiplying the result by A.

Snd. As the line s is generally contained in a plane per-
jiendicular to tlie axis of rotation, and is given in kind, as
well as in position witii respect to this axis, being also gene-
rally symmetrically placed with respect to it, its required
moment of inertia may, in most cases, be most readily
obtained by finding the moment of inertia of a separately,
with respect to two rectangular axes contained in its plane,
and taken through the point m which the given axis of rotation
pierces this plane, and then adding these two moments.

The moment of inertia of a line taken in this way with
respect to a point in its plane has been called by some
writers the wfow moment of inertia.

This method is also equally applicable to finding the mo-
ment of inertia of a plane thin disk revolving around an axis
pei-pendicular to its plane, and to solids which can be divided
into equal laminte by planea passed perpendicular to the axis
of rotation.

(«') The moment of inertia, of the arc of a parabola with
respect to am. ams perp&iMcvla/r to the picc/te of the owve at
a gvoenpoird on the ams of the curve.

Let BAG be the given arc ; A the vertex of the parabola •

./Google

EDITOEIAL APPENDIX.

R tlie point on its axis at wliieli tlie
axis is taken. Throiigh R draw
the chord PQ. Represent (]ie
D eliord EC of the given arc by i ;
its corresponding abscissa AD by a;
and AR by c. Let y represent the
ordinate ?^! and 3! the correspond-
ing abscissa of any element dz of me arc.

^rom the preceding reniai'bs, the moment of inertia of de
with respect to the axis AD will be expressed by j/tfe ; and
that of tlie entire ai'c BAG by

2.y/<fo* = I fy'{]>' + ^^ayt'dy;

as fi'om tlie equation of the parabola, y'=^ 7- a;
By iiitegi'ationf

I,= jj f(J'+ GiaYfdy^l-'^—^ L

ui,wliich Z is the length of the arc BAG.

In like maiiDer the moment of inertia of ds with
to the chord VQ is

f{c-io)'d3
and for the ciiLirc arc BAG,

4,643* '
.pect

,= a/(„-

xfds

4/(-

Sao , 16rt

!,•)(}• J- f;4«y,''%

which intog

■ated as above,

^ 32 . 64 c'o

,)..-

(^4

J-

.\ (f + 16

<o»

" Churo

,'s Ir.t.

ill. An. 199

f Ibid. Art. 160.

./Google

APPEBTDIX;.

From the preceding remarks, the moment of inertia of Z,
with respect to the axis at the point E perpendicular to tlio
plane ot Z, is

"(i + iei?) ki-m--^

The value of Z in the above expressions is

z= i (S- + 16.-)*+ I log., ^-i; + \ s/v + mi ) .•

Each of tlie preceding expressions may be aimplifled, and
an approximate value obtained, sufiieiently near for practical
applications, when the ratios of J and o to a are given. For
example, when "h l_\a there obtains

Z=2a + ^ log. -J- ;

the terms omitfed, being small fractions with respect to
uniw, do not materially affect the result.

Having found the moment of inertia of a parabolic curve,
that of a parabolic i-ing of unil'orui cross section, taken per-
pendicular to the direction of tlie carve at any point, and
having its centre of gravity at its point of intersection with
tlie curve, can be obtained by simply multiplying I,+Ii, by
8 the ai'ea of the given cross section.

(ft') The, moment of msHia of the segrmnt of aparahola with
reject to an ams perpendiciilar to its plane ai a given
point of the cms of the euroe.

Let BAG be the given segment ; A th.e vertex ; AD the

• Church's Int, Cal. Art. 199.

./Google

:, ATPEHDtX.

axis of the curve ; aiid D the point on
the axis with respect to which the mo-
ments of inertia are estiiiiated. Denote
the chord BC by I ; tlie abscisea AD

By (Ai-t. 81.) the momeiit of inertia of
an elementary area pq with respect to AD is Tj{pqfdxt=
^ {^yfdie. That of the segment therefore will be

a ib

I = J^y Sy'cfo = JL y ^ y\l!/ =: ^V ab'.

In like manner the moment of inertia of an clementair
areaasm with respect to the axis BO, is i{j>s)'%=i(a~iK)'
dy. That of the segment Uierefore will be

I.

= \f{a-xfdy = lf{a-'^^^- yjdy = rh\a%

.•.I,-|-I,=^V «S'+ToV f*'^-

From this last expression we readily obtain the moment
of inertia of a disk having the segment for ils base and its
thickness represented by c, with respect to an axis at D per-
^pendicular to its base by simply multiplying I^-l- 1, by e; or

in which -I ale = V, the volume of the disk.

(c') The moment of ifiertia of apwrahoUc dish, or prism,,
■with reject to cm axispa/rallel to the chords which termi-
nate the upper and Imoer iases and midway hepween th^a
chords.

Let pq he an elementary volume of the disk contained
between two planes parallel to the base
EC of the disk. Adopting the same
notation as in the preceding article, the
volume j^j is expressed "by

2y . 0 . dx.

Tlie moment of inertia of this elemen-
tary volume with respect to an axis
through its centre of gravity and parallel

to BC is (Art. 83)

./Google

and tlie moment of inertia of the same volume with respect
to the axis, parallel to the one through its centre of gravity,
taken on the base EC of the disk and midway between the
upper and lower chords is (Art, 79 Eq. 58)

y . 0 . dx {is'-\-{«Mf\ +2;

B {a-xf ;

the moment of inertia of tho entire disk with respect to the
same axis is

.-. I = ■>:, I 2y • 0 . (& jc" + ((&)"{ + / ^ . 0 . dx {a—xj.

Subetitutiug for x and (fo in tei-ms of u, omitting the term
containing (<&)', and integrating as indicated, there obtains,

l^aio (tVs a' + T>^G') = Vm a' + fL<.');

in which V=^af>c.

[d') The moments of inertia of a Tight prism with a, ■brwpe-
zoidal T)aS6 vnih respe^A to aises perpemlioulaT wnd parallel
to the hose at the middle point <f the face terminated hy t/te

Let AGHC be the trapezoid forming the base of the prism.
Eepresent the altitude EF of the trape-
zoid by a ; AG by b ; OH by S' ; and tho
height GB of the prism bye. Let ot be an
elementary volume of the prism between
two planes parallel to the face AB and
at a distance Ee=a! from the face CD.
■ From C drawing Oc parallel to HG there
obtains

E^

'ef ■

:(^-5');

^{5-&') + 6'.

./Google

. APPEHDIX.

The elementary volume^ is therefore

|!(s-f) + y|.

The moment of inertia oip^ '^'-^ respect to an axis throngh
its centre of gravity and perpendicular to the base of the
prism is (Art. 83).

^ij J ^ (5 _ J') + 5' lo.d3)\ {dxy+ ^ (6 - J')+ i' I

and that of the entire prism with respect to an axis at F, the
middle point of AG-, and parallel to the preceding axis, is

omitting the tenn containing {dx)% and integi-ating, aa indi-
cated, there obtains

t b+V

1 which V=a

l + i'

By a like series of operations the moment of inertia of the
entire prism, with respect to an axis pcrpendiculai- to the
preceding one at its middle point between the upper and
lower bases of the prism, will be

T , 17- = i i + Sh' c' ]

, Google

EDITOEIAL APPENDIX.

(e) The inoment of inertia of a "nght prismoid vnth rectcm-
gvla/r hoses wUh, reject to an axis XY through the centre
of gravity of tJie lower hose amd parallel to one of its
eidis.

Let AB — 5, BO = u be the sides of the rectangle of the
lower base ; ab = S', ie = c' the sides
-f of the upper base. Let^^s be any

-j\ Bection of the priemoid parallel to

\\ the lower base and at a distance »

Vr^ from it; and let a be the altitude of

n the priemoid, or the distance between

*\\ its upper and lower bases.

- \-\o From tlie relations between the

.__ V _SY dimensions of the prismoid there ob-

' tains (Art. a')

^/j 7l^ , I. ^ (^ — ^') + ai' .
pq— -{J_5'l-[-5'= ^ — ~,

X , „ , mic — e')+ac'

s^ = -^{<' -<>)+<' = ^ i

and to express the elementaiy solid contained between two
planes parallel to the base of the prismoid and at the height
X above it,

a; (h — i') + <^^ x{c~ c') + ao'

The moment of inertia of this solid, with respect to a
axis fny through its centre of gravity and parallel to XT, i
(Art. 83)

x{h — b') + a/)' x{o — o')+ ao'

- . dx

.{(ifc

The moment of inertia of the prismoid (Art. 79 Eq. 58)

, = A/'-<'-';'+-''x'''-'.'+"-j (fci>±£f-)V m- 1

, Google

HDITOEIAL ArrENDIX,

J a a

omitting the term containing (tfo))' and integrating ae incb
cated, there obtains,

jj5(i5'(c+2cV + S(!o'+4c').

By integrating the expression for the elementaiy Tolume
between the same limits, there obtains to express the volume
of the prismoid

which is the formula usually given in mensuration.

In each of the preceding examples, the quantities I, \, &c,
are expressed only in terms of certain hnear dimensions ; to
obtain therefore the momenta of inertia proper these results
mnst be multiplied by the qnautifcy -, or the unit of mass

con'esponding to the unit of volume, m which t^ represents
the weight of the imit of volume of the material and
^ = 32^ feet.

Each of the above values of I may be placed under more
simple fornis for the gi-eater readiness of numerical calcular
tion by thi'owing out such terras as will visibly affect the
result m only a slight degree. But as such omissions depend
upon the mimericS relations of the linear dimensions ot the
pai-ts no rule for making them can be laid down which will
he applicable to all cases.

( f "j The moment of meriia of a trip hammer.

These hammers consist of a head of iron of which A repre-
sents a side and A a
front elevation; of a
handle of wood B,
which is either of the
shape of a rectaugulai
or of

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EDITOKIAL APPENDIS. 599

two rectangular prlsraoids, having a common base at tlie
axis of rotation C where the tnmnions, npon which the
hammer revolves, are connected flrmly witli the handle by
an iron collar. Another iron collar is placed at the end of
the handle, and is acted on by that piece of the mechanism
which causes the hammer to rotate.

To obtain the moment of inertia of the whole, that of each
part with respect to the axis is eepai'ately estimated and
the sum then taken.

The head A, A' may be regarded as a parallelopiped of
which the side A', reduced to its equivalent rectangle by
drawing two lines parallel to the vertical line that bisects
the figiire, is the end, and the breadth of the side A is the
length. If then from the moment of inertia of tliis parallel-
opiped that of the void a, or eye of the hanimei-, which is
also a pai-allelopiped, bo taken, the difference will be the
moment of inertia of tJie solid portion of the head. The
moments of inei-tia of th^e pai'aUelopipeds may be calcu-
lated, witli respect to the axis 0, by liret estimating them
with respect to the axes tlirongh their respective centi-es of
gravities G and ff, parallel to 0, by {Art. 83) and then with
respect to C by (Art. T9. Eq. 58). Or if the moments of
inertia with respect to Gr and o are small with respect to the
product of their volumes and the squares of the distances
GO and ^0, then the difference of the latter products may
be taken as the approximate value.

The moment of inertia of the handle, if also a parallelo-
piped, will be found with respect to C by {Arts. 79, 83). If
it is composed of two rectangular prismoids, tlien the mo-
ment of the pai'te ou each side of the axis must be found by
(«') and their sum taken.

The moment of inertia of the tnmnions and the iron hoop
to which they are attached may be found by {Arts. 85, 87)
and tlieir sum taken. But as this quantity will be generally
small with respect to the others it may be omitted.

Tliat of the noop at the end of the handle may be taken
approximately as equal to the product of its volume and the
square of the distance between the axis tlirough its ccnti-e
of gravity and that of rotation.

(g') The moment of inertia of a cast iron wheel.
These wheels usually consist of an exterior lim A A <

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EDITORIAL APPENDIX.

uniform cross section coa-
nected with the hoss, or nave
C, 0', which is a hollow
cjlindei', by radial pieces, or
arnas B, B', the cross section
of which is in the form of a
cross. Each arm having the
same breadths at top and
bottom in the direction of
the axis of the wheel as those of the rim and nave which it
connects; the thickness perpendicular to the axis being
uniform. T)ie projection or ribs on the side of each arm,
and whiehgive tlie cross form to the section, being of uni-
form breadth and thickness ; or else of 'unifonn thickness
but tapering in breadth from the nave to tlie rim. Theee
ribs join another of the same thickne^ that projects from the
inner surface of the rim.

Kepresent by K the mean radius of the rim, estimated from
the axis to the centi-e of gravity of its ero^ section ; 5 its
breadth, and d its mean tluckness ; Fits volume, and I its
moment of inertia with respect to the axis; p tlie weight of
'ts unit of volume, aiid ^=32^ feet; then by (Art, 86)

omitting JcT as but a small fractional part of H".

Eepresenting by &, the breadth of tlie arm at the axis,
supposing it prolonged to this line ; 5, its breadth at the rim,
supposing it prolonged also to the mean circle of the rim, d,
its thickness ; F, its volume ; I, its moment of inertia, then

ij («■)

F=E(?.^

and I, :

StTJi--

Representing by a, the breadth at bottom, «, the breadth at
top of the ribs, or projections on the sides of each arm, esti-
mated also at the axis and mean circle of the rim ; d^ their
thickness ; V, their volume ; I, their moment of inertia ;
then by («*')

V,='Rd,'^^' ancll,=

'e K^'

i, + a,

-' + i'i

E"

The sum I-f-I.+Ij will be the moment of inertia of the

./Google

EDITORIAL APPEKDrX. 601

entire wheel approximately, since the moment of inertia of
the portions ot the boss between the arms is omitted, this
being compensated for by supposing the arms prolonged to
tlie axis and to the mean cii'de of the rim. As the qnanti-
ties y„ y,, I, and I, are taken but for one arm, they maat
he multiplied by the number of arms to have the entire
moment.

[h') The moment of meHia of a cast iron steam m^i-ne learn.

These bean^ usually consist of two equal arms symmetri-
cal with respect to a
line a, a' through the
axis of I'otation o.
Each arm, a V a' and
ah a', consists of a
pai"abolic disk of uniform thickness ; S and V being the ver-
tices of the exterior botmding curves, a a' their common
chord, and 06, ob' their axes. The disk is terminated on the
exterior by a fianch B of uniform breadth and thickness. A
lib 0, either of uniform breadth and thickness, or else of
uniform thicloiess, and tapering in breadth from the centre 0
to the ends i, h', projects from each face of the plane disk
along the axis 5 o. The beam is perforated at the eenti-e,
near the two extremities and at intermediate points, to
receive the short shafts, or centres aromid which rotation
takes place. Around each of these peiforations, projections,
or bosses D', D", &c., are cast, to add strength and give a
more secure fastening for the shafts.

The beam being symmetrical with respect to a a', it will
be only necessary to calculate the moments of inertia of the
component parts of each arm with respect to tlie axis o and
take double their sum for the total moment of inertia of the
beam. Those component parts are — let, the parabolic flanch ;
2nd, the parabolic disk of unifonn thickness enclosed by the
flanch ; 3d, tlie rib on each side of the disk, running along
the centi'al line hh' ; 4th, the projections, or bosses I)' &c.,
around the centres.

The moment of inertia of the flanch will be calculated by
(«') as its thickness is small compared with the other linear
dimensions. That of the disk will be calculated by (6').
That of the rib by (««'). Those of the projections may be
obtained within a sulhcicnt degi'ee of approximation by

./Google

602 EDITOEIAL APPENDIX.

taking tile product of their volumes and tlie sijnares of theii
respective distances from the axis o.
The sum of tlieee quantities being taken it must he multi-

pHed hy - as in the preceding cases ; h- heing the weiglit of

the unit of volume of the mateiial.

Note (a).
The increase of tension due to rigidity and wliicii is f

. P, _ c^ (a + & ■ F,)

by writing e™ . « for D, and c™ . t for E, in which o repre-
sents the circumference of tlie rope, and m the power to
which e is raised.

The increase of tension of any other rope whose circumfer-
ence is Cj bent over the same pulley and subjected to tlie
same tension P, is, in like manner, expressed by

c.'"(a+5F.)

ITow representing by T and T, the two values above for the ,
respective increase of tension for o and c^ there obtains, by
dividing the one hy the other,

which expresses the rule given above for using the tables in
ealcnlating the increase of rigidity due to a cord whose cir-
cumference is different from those in the tables.

IS'OTE it).

A.S one of the chief ends of every machine designed for
industrial purposes is, under certain restrictions as to the

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EDITOKIAL

quality, to }'ield the greatest amount of its products for tlie
motive power consumed, it "becomes a subject of prime
importance to see cleaidy in wliat way the work yielded hj
tile motive power to the receiv^er, at its applied point, is
dimiuiehed by the various prejudicial reaiatanceB, in its
transmission through the material elements of the machine
to the operator, or tool by whicii the products in question
are formed.

The most convenient method for doing this will be to
place (equation 112, Art. 145) which expresses the relation
between the work ^U^ of the motive power at thfe applied
point and that 2"D"j the work of the operator at tlie working
point, with the portion 2U+ — Sw {^^—v^) which repre-

sents the work consumed by the prejudicial resistances and
the inertia, under a form such that the work of each preju-
dicial resistance shall be separately exhibited, for the pur-
pose of deducing, from this new form of the equation, the
influence which each of these has in diminishing the work
yielded at the applied point and transmitted to the operator.
To effect this change of foiTQ in (equation 113) designate by
Pi the motive power, and Si the path passed over by its
point of application in its line of direction between any two
intervals of time, during which P, may be regarded as vari-
able both in intensity and direction ; P, and Sj the resistance
and corresponding path at the working point ; E the various
prejudicial resistances which, like fnction, the stiffness of
cordage, &c., act with a constant intensity, or are propor-
tional to Pi, and S tlieir path ; w^ the weight of the parts the
centre of gravity of which has changed its level during the

period considered, and H its path ; and — 'w{v,'—Vi')~im,

{v^—v^) the half of the difference between the living forces
or the accumulated work of the material elements in motion,

of which m = — is the mass, during the same period, in

which tlie velocity has changed from v, to v,.

Now for an elementaiy period di of time, duiing which
the forces P, &c., may be regai'ded as constant, and their
points of application to have described the elementary paths
rfSi &c., in tlieir lines of direction, (equation 113) will take
the form.

./Google

(i04 EDIT0EI4L APPEHDIX.

in which tlic 1st member of the ef[uation expreasea tlio inere-
ment of the Hying force, or the elementary accumulated
work for tlie interval dt at any instaut when tlie velocity of
tlie mass m is i>; and the 2nd member the corresponding
algebraic sum of the elementary work of P„ E, &c. This
equation being integrated between the limits ^, and t^ in
which ■!) changes from v^ to w, there obtains,

■ ■ (J3).

./.

Tliis equation (E) is the same as (equation H2), The
symbol s designating the aggi'egate of the work of the

various forces of the same kind ; and that as / P,(^, &c.
the work of each force as P„ supposing it to be either con-
stant or variable. In either case wlienerer P, &c., can be
expressed in terms of S, the value of / Pifl^S, can be found
by one of the methods in (Notes I and m)\ and suppoBing P,
&c., to represent their mean values, and S, &c., the paths
described in their true directions during the interval con-
sidered, equation (B) may be written under the following foiin
for the convenience of discussion,

^(i,^'_i,^')=P,S -ES-PA±WH (C).

In tills last equation 2 / w, c?A=WH (Art. 60) represents
the work of the total weight of the parts whose centre of
gravity lias changed its level during the iutei-val considered,
and it takes the double sign ±, as the path H may be
described either in the same, or a contrary direction to that
in which W always acts.

Before procee<nng to discuss the terms of (equation C),
it may be well to remark that the tei-m — ES does not take
into account the work expended by P^ in overcoming the
molecular forces brought into play by the deflection, torsion,
extension, &c., of the parts of the machine ; for, owing to
the rigidity of these parts, this forms but a very small trac-
tional part of the total work of the exterior forces whilst the
maclune operates eontuiuously for some time; as, dui-iug

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EDITORIAL APPENDIX, 605

this time, tlio tension of the pai-te, or the iiioleciilar resist
aiTces remain sensibly the same, and the molecular displace
ments are for the most part inappreciable, or else very small
compared with the paths described by the points of applica-
tion of the other forces.

This remark, however, does not apply to the expenditure
of work by the motive power where the operation of the
machine requires that some of the parts in motion shall be
brought into contact with others which are either at rest, or
moving with a slower velocity so as to produce a shock-
In this case there may bo a very appreciable amount of
living force, or accumulated work destroyed by the shock,
omng to the constitution of the material of which the parts
are composed where the shock takes place ; and, if tlie shocks
are frequent during the interval considered, and in which
the other forces continue to act, tho accumulated work
destroyed during this interval may form a large portion of
tlie work expended, or to be supplied by the motive powei-.
In calculating tliis amount of accumulated work destroyed,
we admit what is in fact true in such machines, that the
interval' in wMch the shock takes place is infinitely small
compared with the interval in which the other forces act
continuously, and therefore, in estimating the accumulated
work destroyed in each shock, that we can leave out of
account tlie work of the other forces during this infinitely-
small interval. In this way, considering also that tlie parts
where the shock takes place arc usually formed of materials
which undergo an almost inappreciable change of form fi-om
the shock, and that therefore the mechanical combinations
of the machine are sensibly the same after the shock as
before it, we readily see that, to obtain the total expenditui-e
of work by the motive power, for any finite interval, we must
calculate that which is consumed by all the other resistances
during this inteival, and add to this that destroyed by the
shocks during the same interval, the latter being calculated
irrespective of the work of the other forces during tlie short
duration in which each shock occurs.

"We thua see that, except in some cases where tlie great
velocity of the paiis in motion may give rise to an appreci-
able expenditure of work caused by the resistance of the
medium in which these parts may be moving, as the air, &c.,
the forces which act upon any machine in motion are the
motive power ; the resistances, such as friction, stiffness of
cordage, &c., which act either with a constant intensity
dm'iug the motion, or arc proportional to the motive power ,

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the weight of til e parts wliose centres of gravity do not remain
on the same level d^lril^g this interval ; the useful resistance
arising from the mechanical functions the macliine ia designed
for ; and the forces of inertia which either give rise to accu-
mulated work, or the reverse, as tlie velocity increases, or
decveasee during the interval considered.

Tteenming equation (0) we obtain, by transposition,

That is the useful work, or tliat yielded af; the working point
and which it is generally the oWect of the machine to make
as great aa possible consistently with the quality of the
required products, will be the greater as tlie tei-ms in the
second member of the equation affected with the negative
sign are the smaller.

Taking the teriii — ES, it is apparent that all that can be
done is to endeavor in the case of each machine to give
snch forms, dimensions and velocities to those parts where
these resistances are developed as will make it tlie least

With respect to "WH it will entirely disappear from the
equation when H=o ; in which case the centre of gravity of
the entire' system will remain at the same level; or else
only tliat poition of this term will disappear which belongs
to those parts of the machine whose centres of gi-avity either
remain at rest, as in the case of wheels txictly centeied, end
less bands and chains, &c. ; or m the case of thoi^e pieces
■which receive a motion simply m a hoiizontal diieetion
This term will also disappear in whole or in put, m thoe
cases where tlie centre of gravitj ascends and descends
exactly the same vertical distance m tlie interval correspond
ing to the work P,S, ; for during the ascent, as the direction
of the path H is opposite to that of the weight W, the work
consumed will be — WH, whereas, in the descent, it will
restore the same amount or +WH, and the sum of the two
will therefore be 0. This takes places in the parts of many
machines, for example in crank arms, and in wheels which
are not accurately centered ; in both of- which cases the
centi'c of gravity ascends and descends the same distance
vertically in the interval coiTesponding to each revolution
of tliese parts whilst in motion ; also in those parts of a ma-
cliine, like the saw and its frame in the saw mJU, which rise
and fall alternately the same distance.

In aU of these cases then the useful work P„S, will not be

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. APPENDIX.

affected by tlio work doe to the weight of the parts in
question.

It may \>6 well to observe that the preceding remm-ks refer
only to the direct influence of tlie weight of the parts on the
amoant of useful work ; but whilst directly it may produce
no effect however great its amount, the weight, indirectly,
may cause a considerable diminution of tiiis work, by
increasing the passive resistances and thi"a the teiin ES.
The same holds with regard to the accumulated work, repre-
sented hy the term ^iv', from whicli a considerable dimi-
nution may be made in P,S, if this accumulated work cannot
be converted into useful work, and thus be made to form a
portion of P,S„ when the action of the motive poweris either
withdrawn, or ceases, by variations in its intensity, to yield
an amount of work which shall suffice for the work consumed
by the resistances.

These last remarks naturally lead us to tlie consideration
of the two tenns imo', and —^mv', or half the living forces,
or accumulated work at the commencement and end of the
interval considered. As the machine necessarily stai'ts from
a state of rest under the action of the motive power P„ it
follows that imv', the accumulated work due to tliis action
tends to increase P^S,, whilst that —^iivo^ is so much accu-
mulated in the moving parts by which PjS, is lessened.
This diminntion of P^^ is but inconsiderable in comparison
with the total useful work when the interval in question, and
duiing which the machine operates without inteniiission, is
great ; also in cases where the velocity attained by the parts
m motion is inconsiderable, as for example in machines em-
ployed for raising heavy weights, in which ^iiv^ will in
most cases be but a smaU fraction of the useful work which
is the product of the weight raised and the vertical height it
passes throngh. In this last example we also see the incon-
veniences which would result from allowing bodies raised by
machinery to acquire any considerable amount of velocity ;
or to quit the machine with any acquired velocity, as, in
this case, the accumulated work generally would be entirely
lost so far as the required useful effect is concerned.

Except in the case where the accumulated work ^m/w,'
can be usefully employed in continuing the motion of the
machine and gradually bringing it to a state of rest when the
motive power P, has either ceased to act, or has so far
decreased in intensity as to be incapable of overcoming the
resistances, whatever tends to any augnientation of living
force should be avoided, for the teim which represents tliis

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608 JSPITOEIAL APPENDIX.

teing composed of two factors the one representing the mass
of the parte iti motion and the other tlie square of its velo-
city, it is evident that the prejudicial reeistaaces eueh as
Iriction on the one hand and fiie resistance of the air on the
other will increase as either of these factors is increased, and
thus a verj appreciahie amount of this accumulated work
may be consumed in useless work caused hy the veiy in-
crease in question. If, moreover, the machine fi'om the nature
of its operations is one that requires to be brought suddenly
to a state of rest, any considerable amount of accumulated
work might so increase the effects of shocks at the points of
articulation as to endanger the safety of the parts.

The foregoing remarks apply only to those paiis of a mar
chine where the direction of motion remains the same whilst
tlie macliine is in operation. Where any of the parts have
a reciprocating motion, in which case whilst the part is
moving in one direction the velocity increases from 0 up to
a cei-tain limit and then decreases imtil it again become 0
at the moment when the change in the du'ection of motion
takes place, and so on for each period of change, it will be
I'eadily seen that where the velocity varies by insensible
degrees, the accumulated work of these parts foi- each period
of change will be 0 and will therefore have no influence on
the amount P^Sj of ueefid work.

The avoidance of abrupt changes of velocity in any of the
parts of a machine is of great importance. The mechanism
tlierefore should, as a general rule, be so contiived that there
shall be the least play possible at the articulations of the
various parts, and that the articulations shall receive such
forms as to procure a continuous motion. In cases also
where any of the parts have a reciprocating motion such
mechanical contrivances shonld be used as will cause the
variations of velocity in tliese parts, within the range of
their paths, to talre place in a very gradual manner ; such
for examples as what obtains in the cranks and eccentrics
■which are mostly employed to convert the continuous circu-
lar motion of one pai"t into reciprocating motion in another,
or the reverse.

There are some industrial operations however which are
performed by eliocks, as in stamping machines, trip ham-
mers, &c., and in these cases the useful work is due to the
work developed by the motive power in raising tlie pestle
of the stamping machine, or the head of the trip hammer
through a certain vertical distance from which it again falls
upon the matter to be acted on, having acquired in its

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EDITOEIAL APPEHDIX. 609

descent an amount of living force, or accuraiilated work due
to the height through which it has heen raised. In such
caees it is to he noted that, independently of the wort due
to the motive power consumed by the resistances whilst the
hammer or pestle is kept in motion by the other pai-ts of the
mechaniem, and wliicli is so mnch uselessly consumed so far
ae the useftil work is eoueei'ned, there wUl he a portion of
the accumulated work in the pestle, or hammer also uselessly
consumed, arising from the want of perfect rigidity and
elasticity in the material of which tliese two pieces are
usually composed. Brides this, both tlie pestle aiid matter
acted on may and generally do nave relative velocities after
the shock between tliem, which as they are foreign to the
purpose of the operation, will also represent au amomit of
accumulated work lost to the useful work. Prom this we
may infer that, as a general rule, other industrial modes of
operating a change ot form in matter will be preferable to
those by sliocla, whenever they cau he employed ; and that
such modes are moreover advantageous, as they avoid those
iars to the entire mechanism which accompany abrupt
changes in the velocity of any of the parts, and which, by
loosening tlie articulations more and more, increase the evil,,
and ultimately rendei- the machine unfit tor service.

Having examined the influence of all the various hurtful
resistances brou^t into action in the motion of machines,
upon the work PiSi expended by the motive power, and

fointed out generally how the consumption of the work may
e lessened, and the useful work to the same extent increased,,
we readily infer that like observations are applicable to the
term P^S, the work of the r^istance at the working point.
As the prime object in all industrial operations performed by
machinery is to produce the greatest result of a certain kind,
for the amount of work expended by the motive power, it
will be necessary to this end that the velocity, the foi"m, &c.,.
of the operator, or tool by which tlie result sought is to be
obtained, should be such as will not cause any useless expen-
diture of work. On this point experiment has shown that
for certain operators there is a certain velocity of motion
by which the result produced will be tlie mt^t advantageous
both as to the quality and quantity.

With respect to the work of the motive power iteelf repre-
sented by the product PjS, it admits of a maximum value ;
for when the receiver to which P^ is applied is at rest, P, will
act witli its OT'eatest intensity, but the velocity then being 0
the product P,Si will also be 0 ; but as the velocity increases

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610 EDnOKIAL APPENDIX.

after the receiver begins to move the intensity of the action
of Fj upon it decreases, i]ntil finally the velocity of the
applied point may receive such a value V that P, will become
0, and the product P,S, in this case will then also be 0. Ab
the -work P^S, thus becomes 0 in these two states of the velo-
city, it is evident that there is a certain value of the velo-
city which will make PjS, a maximum. To attain tliia
maximum the mode of action of the motive power selected
on each form of receivei' to which it is applicable will require
to be studied, and such an aiTangement of its mechanism
adopted aa will prevent any decompositions of the motive
power tliat would tend in any manner to increase the hurt-
ful resistances and thus diminish the useful work.

It will he very easy to show that the laws of motion of all
machines, that is the relations between the times, spaces and
velocities of the motion of any one of the moving parts are
implicitly contained in the genei'al equation of living forces
as apphed to machines which has just been discussed.
Eeauming (equation B) with this view, and representing by
dm any elementary mass in motion whose velocity is «, at
any instant when it has described the path, or space a, if we
take any other elementary mass Bm, in a given position and
denote by v, its velocity at the same instant, we shall have
(;,=M, (ips), and u,=Mi (f^i); in which ?« is a purely geome-
trical function, since, from the connection of tlie parts of a
machine, in which any motion given to one pai't is trans-
mitted in an invariable manner to the other, the space passed
over by any one point can always be expressed in terms of
that passed over by any other assumed at pleasure.

From the relations v^^u^ (tpfi), and i), dt^ds, we obtain

u,'^(^s'f=v' and u^du, {^sf^i\dti^-= -j-^ds.

Substituting these values of v' and «, d/o, in (equations B and
A), and letting in still represent the sum of the elementary

^masses as dm, there obtam the two equations
:^fv.,d^,±^fwdh. (B')

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EDITOKIAL APPENDIX. till

sP,dS,±2w(ZA. (A'),

the livBt showina; tKe relations between any two states of the
velocities u, and m, for any dofinite interval, and the second
for the intinitely BmaU interval cfo. Now as tlie i-elationg
hetween the quantities rfS,, dS„ &c., or the elementary
patlis described by the points of application of P„ P,, &c.,
and the elementaiy apace cfo, from the connection of the
parts of the machine, can be expressed in functions of s and
of the constants that determine the relative magnitudes and
positions of those parts ; and as, moreover, P„ P^ &c., are
either constant, or vary according to certain laws by which
they are given in functions of the patha S^ S„ &e., we see
tiiat aU the relations in question are implicitly contained in
the two preceding equations.

Let us examine the kinds of motion of which a machine is
susceptible and the conditions attendant upon them. We
observe, in the first place, supposing &e machine to start
from a state of rest, that the elementary work P,dS, of the
motive power must he greater than that of the resistances
combined, or P,i^S,— Ec^S— &c. >0, so long as the velocity
is on the increase. The living force is thus increased at
each instant by a quantity d {7n/^=2mvd^, or by an amount
which is equal to twice the elementary work ot the motive
power and resistances combined: and tliis increase will go
on so long as the elementary work of the motive power is
greater than that of the resistances. But, fi'om the very
natm-e of the question, this increase cannot go on indefinitely,
for the point of apphcation of the motive power would in the
end acquire a velocity so great that P, would exert no effort
on the receiver, whereas the resistances still act as at the
commencement, and some of them even increase in intensity
with the velocity. The living force tlierefore will, at some
period of the motion, attain a limit beyond which it will not
mcrease, a fact which the operation of all known machines
confirms, and, having thus reached tliis state, it must either
continue the same during the remainder of the Jime that the
machine continues in motion, or else it must commence to
decrease until the velocity attains some inferior limit from
which it will again commence to increase, and so on for each
successive period of motion during which the action of the
forces remains the same.

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613 EDITOEIAL APPENDIX.

Supposing the madiine to continue its motion with the velo-
city it has attained at this maximum state of the Uving force,
we shall tJien have

and

inasmuch as the motion being now uniform the difference
between the living forces corresponding to any finite inter-
val of time is 0, Consideriug the manner in which the parts
of machines are combined to transmit motion from point to
point, we infer that this condition with respect to the
incre^e of living force, and which constitutes uniform
motion, can only obtain when the velocities of all the differ-
ent parts bear a constant ratio to each other. Kepresenting
by v', v", v'", &c,, these velocities which are respectively

equal to -^r-, ——, -4-i ^c-i we see that the ratios of ds',

^ di dt dt

ds", da'", &c., will also be constant when those of -y', v", &c.,
are so ; that is, this constancy of the ratio of the effective velo-
cities and of the quantities ds", ds", &c., must subsist together
for all positions of the parts of machines to which tliey refer ;
but as the latter, which are the virtual velocities, or ele-
mentary paths described, depend entirely on the geometrical
laws that govern the motion of the parts, a httle considerfttion
©f the various mechanical combinations by which motion is
transmitted will show that, in order tliat their ratios shall
respectively remain constant, no pieces having a reciprocat-
ing motion can enter into the composition of the machine,
as the velocities of such pieces evidently cannot he made to
bear a constant ratio to tlie othera. Tina condition it will be
seen refers exclusively to the mechanism of the machine, or
the geometrical conditions by which the parts are connected,

and nas nothing to do with the action of the forces them-

selve-B.

But when the condition of nniform motion is satisfied

there obtains also

7,dS.-'RdS-'P,dS,±yfdIl=Q ;

that is, according to the principle of virtual velocities, an
equilibrium obtains between tlie forces which act on the
machine irrespective of the inertia of the parts. As a gene-
ral rule this condition required that not only must the forces

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APPENDIX. G13

P„ K, ifec, be constant both in intensity and direction and
aet continuously, but that the term WiMl must be sepa-
rately equal to 0, or the centre of gi'avity of eacli part must
preserve the same level during tne motion ; for were tliis
not so any piece whose weight is w would evidently impress
an elementary work represented by ±wdh wliich would be
variable in the different positions of the mechanism ; unless
w, having itself a uniform velocity, formed, as might be the
case, a part of the motive power P„ pr of the useful resist-
ance P,.

It thus appears that to obtain uniform motion not only
must tlie mechanism used for transmitting the motion con-
tain no reciprocating pieces, and therefore consist solely
of rotating parts, as wheels, &c., and parts moving continu-
ously in the same direction, as endless bands, and chains, &c. ;
but that the centres of gravity of these pieces shall remain
at the same level during the motion, which will require that
the wheels and other rotating pieces shall be accurately cen-
tered so as to turn truly about their axes.

The diificulty of obtaining a strictly uniform motion in
machines is thus apparent, for it involves conditions in them-
selves practically unattainable, that is, applied forces acting
contuiuously and with a constant intensity and direction, and
that the ratio of the virtual velocities of tlie different pai-ts
should be constant and independent of the positions ot the
mechanism, a condition which requires that the terms (tp*)
and 2dm{<psy in the preceding equations shall also be con.
stant for all of these positions. But even were these condi-
tions satisfied, it can be shown that rigorously speaking a
machine starting from a state of rest will attain a uniform
velocity only in a time infinitely great. This will appear
from geometrical considerations of a very simple chai-acter,
or from the fonn taken by equation. By the first method,
let OT, OY be two co-ordinate
axes, along the one set off the

V abscissas 0:f', Oif', &c., to re-

j^J^T""' present the times elapsed from

y\ '• the commencement of the mo-

/ ! I tion, and the ordinates fv', t"v",

I , h—^ T — &c., the corresponding veloci-

ties, the curve Ov'v", &c., will
give the relation between the times and the velocities, Now,
from the circumstances of the motion, the increments of tiie
velocities will continually decrease, and the curve, from the
law of continuity, will approach more nearly to a right line

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61i EorroitiAi. appehdix.

as the time increases ; having for its assymptote a right line
parallel to OT, drawu at a distance Ov from it, whieli is the
limit the velocity attains when the motion becomes uniform.
We moreover see from the form the curve may assume that
this limit will be approached more or less rapidly.

From (equation B'), representing by c the quantity
'£m{(p8f, we obtain

at m as as as

Now, from the preceding discussion, the forces being sup-
posed to act continuously, and with a constant intensity and

direction, and the quantities — ^^ -=-' being constant, the

as ^ as
function expressed by the second member of this equation
has its greatest value when 'ij,=0, or when the machine is
about to move, and that after motion begins it decreaees
more or less rapidly as the velocity increases, until it be-
comes 0 for a certain finite value of the velocity. Hence it
follows that the function must be of the following, or some
equivalent form,

in which J is essentially positive and a function of «, and
certain constants, and V is the limit of the velocity in ques-
tion. "We shall therefore obtain from (equation B'), by sub-
stituting tliis function for the second member.

The second member of this last equation, when integrated
between the limits '!'j=0, and «j=^ v , must contain, according
to the known rules applicable to it, at least one terra of the
form of — a log. (V — «j) if the exponent n is odd ; or
— a(V— 'y,)'"-'-^ , if n is even ; either of which functions will
become infinite for Y— r5=0, or when v, attains its limit.

From the conditions requisite to attain uniformity of mo-
tion in a machine, the advantages attendant upon it, so fai'
as it affects the mechanism are self-appaa-ent ; not only will
there be none of that jarring which attends abrupt transi-
tions in the velocity, but, from the manner in wliich the

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EDITORIAL APPENDIX. bJ.5

fttrces act, the strains on all the parts wiH he ec^uable, and
the respective form and strength of each can thus be regji-
lated in aecoi-dance with the strain to he brought upon it,
thus reducing the bnlk and weight of each, to what is strictly
req^nisite for the safety of the machine. But advantages not
less important than these result from the use of mechanism
ansceptihle of uniform motion, omng to the fact that for
each receiver and operator there is a velocity for the applied
and working points with which the functions of the machine
are best performed as respects the products ; and these
respective velocities can Se readily secured in unifoiin
motion by a suitable arrangement of the mechanism inter-
mediate between those two pieces.

The advantages resulting from uniform niotion in machines
has led to the abandonment of mechanism that necessarily
causes in'egulai-ity of motion, in many processes where the
cliaracter of tiie operation admits of its being done ; and
where, from the manner in which the motive power acts on
the receiver and ia transmitted to the operator, parts with a
reciprocating motion have to be introduced, every possible
care is taken to so regulate the action of these paiis and to
confine the working velocity within the narrowest limits that
the character of the operation may seem to demand. Many
ingenious contrivances have been resorted to for this pur-
pose, hut as they belong to the descriptive part of mechanism
rather than to the object of this discussion, and, to be undei--
stood, woidd require diagrams and explanations beyond the
limits of this work, they can only be here alluded to. There
is one however of general application, the fiy wheel, the
general theoiy and application of which to one of the sim-
T^est cases of iiTegularity are given in (Arts. T5, 1Q, S65, &c.)
TTie fimctions of this piece ai'e to confine the change of velo-
city, arising from irregularities caused either by the mechan-
ism, or tie mode of action of the motive power within certain
limits ; absorbing, by the resistance offered by its inertia,
or accumulating work whilst the motion is accelerated, and
the work of the motive power is therefore greater than that
of the other resistances; and then yielding it when the reveree
obtains ; thus performing in machineiy like functions to
those of regulating resei-voirs in the distribution of water.
It should however not be lost sight of that whatever resources
the fiy wheel may offer in this respect they are accompanied
with drawbacks, inasmuch as the weight of the wheel, its
hulk and the great velocity with which it is frequently
required to revolve, add considerably to the prejudicial

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resistances, as friction and tlie resistance of the air, and thne
cause a useless consumption of a portion of tlie work of the
motive power. Whenever therefore, by a proper adjnetment
of the motive power and the resistances, and a suitable
arrangement of the mechanism, a sufficient degree of regu-
larity can be attained for the character of the operation, the
use of a flywheel would be injndieious. In cases also where,
from the functions of the machine, its velocity is at times
rapidly diminished, or sudden stoppages are requisite, the
fly wheel might endanger the safety of the machine, or he
liable itself to rupture, it should either be left out, or else
the mass of the material should be concentrated as near as
practicable around the axis of rotation ; tlins supplying the
requisite energy of the fly wheel by an augmentation of its
mass. In all otlier cases the matter should be thrown as far
from the axis as safety will peniiit, as the same end will be
attained with less angmentation of the prejudicial resist-
ances.

From this general discussion some idea may be gathered
of the relations between the work of the power and that of
tlie resistances in machines, and of the means by which the
latter may be so reduced as to secure the gi'eatest amount of
the former being converted into useful wort. It must not
however bo concealed that the problem, as a practical one,
preeents considerable difficulty, and requires, for its satis-
factory solution, a knowledge of the various operator and
receivers of power, as to theu' forms and tlie best modes of
their action. This knowledge it is hardly necessary to
observe must, for the most part, be the result of experiment;
theory serving to point out the best roads for the experi-
menter to follow. Both of these have shown that the work
of the motive power consmned by the resistances, caused by
the parts tlirough which motion is communicated from the
receiver to the operator, is but a small fractional part of the
total work uselessly consumed, whenever tlie mechanism has
been aiTanged with proper attention to the functions required
of it ; but tiiat the principal loss takes place at the receiver
and operator, and iiiis is owing to the difficulty of bo arrang-
ing the receiver that the motive power shall expend upon it
all its work without loss from any cause ; and in like manner
of causing the operator to act in the most advantageous way
upon the resistance opposed to it. Some of the general con-
ditions to which these two pieces must be subjected, as to
unifoi-mity and continuity of action of the motive power and
the resistances, and the avoidance of jarring and ehoeks have

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APPKNDIX. 617

been pointed ont, as well as the fact that to each correspond?
a certain velocity by which the greatest amount of nseftil
effect wiU be attained.

This discussion will make apparent that, comparatively
speaking, but a small amount of the woi-k due to me motive
power is expended on tlie usefn! resistance, or tlie matter to
be operated on. In some of the beat conti'ived reeeivere, as
the water wheel, for example, where the motive power can
be made to act with the greatest regularity, and the receiver
be brought to as near an approach to uniformity of motion
' ' ainable, the quantity of work it is capable of yielding
eiffht tenths of that due to what the v

expends upon it, under the most careful arrangement of the
wheel and the velocity of its motion.

As an example under this head (Art. 149) equation (115),
and an illustration of the cireunistanees attending the attain-
ment of uniformity of motion Kote {t) in machines ; suppose
the axle A eai'rying two arms B, B, to the
extremities of wiiich two thin rectangular
disks C, C, are attached, their planes pass-
ing through the axis of rotation, to be piit
in motion by the descent of a weight P, at^
tached to a cord wound round the axle.
In this case the resistances to the moving
force duiing the accelei-ation will be
that of the air acting against the disks and
the two ar-ms, the inertia of the parts in
motion, and the friction on the gudgeons
of the axle.

Eepreeent by A the sum of the areas of
i, a the distance of their centres from the axis,
. / mass of the machine at the distance r
from the axis, u the angular velocity of the system, a, the
radius of the axle measured to the axis of the cord, p the
radius of tlie gudgeon, if the limiting angle of r^istance,
l^ the total length of the cord, I the length of the part
unwound, w the weight of tlie unit in length of tlie cord, ~W
the total weight of the machine excepting F,.

From experiment we have for the resistance of the air to
the motion of the two discs oA«'=i;A<u''ffl', in which v=<iia

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HIS EDITOKIAL APPENDIX.

expi'eEsea the velocity of the centi'e of the disk and e a eon
etaiit determined by experiment. The resistance offered b^'
the inertia of dm during the acceleration of the motion la
represented (Ai-t. 95} ecLuations (T2) (73) by dmr — -, in

which is the acceleration of the angular velocity in the

at
element of time di, the resistances offered by the inertia of
tlie weight P, and that of the pendant portion of tlie cord
represented by wl are, in hke manner, expressed -.'"'" - «, — — ,
the total pressure upon the gudgeons wiU evidently be ex-
pressed by P.+W— -^^ti£. ffl,-—-, since, during the accele-
^ g ' di' ' ^

ration of the motion, fhe resistance of the inertia of the
weights Pi and wl act in an opposite direction to these
weights.

In the state bordering upon motion at each instant there
obtains

du ^T,+wl^,di
P, + wZ d<^

dt ^

/p , -^7-_F, + ^"^ ^ d<^
\ '"^ a ' di

psni.a

Eepresenting by if the coefficient of (j", by m? that of — — ^
and by / the algebraic sum of the other terms, there obtains

n^d'Jrin'— —d'=0. :. dt=m^- -^'^ .

dt q'—nW

{ I!:'' I

• t=m' f-J!^=!^lo<^ Ji±^\ •<. = ^-^- —■

From this last ecjuation we se that w approaches rapidly the
limit ?- which it only attains when i=QO-. As this limit cor-
responds to that in which the motion would become uni-

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form, it might have teen dednced directly from the first of

Note ('it).

of e&UmaiAm,g the amount qfworii consumed by the
trip hammer.

The trip hammer is used in forging heavy iron work,
motion being given to it for this purpose by teeth, termed

ca ns iim ly fixed m an axle A ter ned he oa « /
omd ■wlu h t y e ai anged a e al te val aj mt
The a 0 he hanme f u n a ed an uon and the

ufi e s ta e of wh ch ece e. s b e to m to wo i.
tia y w th he face t the cnw le oena n

contac d nng he a.cen of he id of the ha ume on the
8 me pnncple as he te h i e nel n oler ca.

The n e val betwe nlea socl edh ch
cam shall take he band it re at tl e p on he ho

zonta 1 ne G C J n g the cen e. of r a n f thts cam
shaf a 1 ha nme
To estimate e w k co sumed n t! e play ot tl s a
h ne t n net be ob e -ved ha t con t ot ee 1 sti ct
pa H the fi t s t a co un ed by e ni a t or shoe
the second that due to the peiiod after the shock, in which
the cam and tail of the hammer remain in contact ; the thwd
that consumed by the cam shaft in the interval between the
separation of the cam and hammer and the moment when
the succeeding cam takes the hammer.

Denote by K, the radius of the primitive circle 0,^ of the
cams ; by w, the angular velocity of the cam shaft at any
period of the shock; by pj the radius of the gudgeon oa

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620 EDiroEi

which the shaft revolvea ; by ffl^ the limiting angle of resist-
ance for the eui'faces of the gudgeon and its bed ; by m an
elementary mass of the shaft ; by r tlie distance of m from
C, ; \)j Il,=C,i, w„ p„ (p, »ij, and r^ the con-eepoiiding quan-
tities for the hammer.

Now if we represent by P the mutual pressure between
the surfaces of tlie cam and band at any period of the impact,
there must be an equilibrium at each instant between P and
the forces of inertia and the passive resistances developed
in the play of the machine. Considering the equilibrium
around the axis of rotation C, of the hammer in the first
place, we have for the velocity of any element TOj, at any
instant, r,w„ and for the increment of velocity impressed
upon it by the cam T,di^, ; the force of inertia therefore deve-
loped by this increment is expressed by

and its moment with respect to the ajcis 0, is

and fbe sum of the moments of all the forces of inertia ia
(Arts. 95, 106)

To obtain the friction on the trunnions of the hammer due

to P and the resultant of the forces of inertia J"-,^,-^, we

dt
have for the resultant of the latter (Art. 108) equation (83)

dt

in which M represents the mass of the hammer, its handle.
&c., and G the distance of its centi-e of gravity from 0, the
axis of rotation. Now, decomposing this resultant into two
components perpendicular and parallel to the line C,C, repre-
senting by a the angle between this line and the one O^G
through the centre of gravity of the hammer, &c., we have
for the perpendicular component

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EDITORIAL APPENDIX.

^ttMG coa. a

and for Lhc parallel one

-^ MG- sin. a.

at

Tlie total pressure on the txEiiiiions, from P and the fovcea
of inertia, will therefore be

/(^

As however, in most cases of practice, tlie angle a is either 0,
or very small, the value of the quantity under the radical
may he taken without sensible error

^'^ dt '"■^'
The equation of equilibriura ahoiit the axis C, is therefore

PIi,=-^M,E.'+ (P + -^-^ra}f>, sin. 9..

t?6), MiE,°+!MG-p,sin.y,
" dt E,— piSin. 9,

. . (A).

Kow with respect to the cam shaft we have, to express the
siun of the moments of the forces of inertia with respect to
the axis 0^,

di di ' "

As the pressure on the trunnions of this shaft is dne to the
force P alone, the moment of the friction on tliera will he
expressed hy P p sin. 9,.

The ecLuation of equilihrium of all the forces with respect
to 0, will therefore he

-^M,E;=^PE,+Pp, sin. 9, (E).

'Kliminatiiig P hetween equations (A) and (B) thero obtains

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C22 EOTTOEIAL APPENDIX.

at ' ' K, — p,Bm.ffl, ^ ' ' " '' dt ^ '

Ihe coefficient of -^ can be M-ritten as follows,
di '

E.,— p, sm. ip, '^ '

,_P,sm. ?i \ M.E," I ' ■" '

placing K for the coefficient of MjE^Ej. MaJiing these sub-
stitutions in equation (C) there obtains

•M.E.'r^^KM^E.E./'' ^'^.
' ^J dt . v_ dt

.: M,E;(n-6.,)=KM,E>,

in ■which U represents the greatest, and Wj the least angular
velocity of the cam shaft ; and ^^=0, &),= -^^} the angu-
lar Telocities of the hammer ; since before the impact it is
at rest, and finally attains the same velocity as the cam has,
in which, from tlie circumstance of the mechanism, w,E,=

From the preceding equation there obtains
<..= ""■ ....(D).

Now, as a general nilc, the quantities '■""'-''■, '■'"■''■ and

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EDnOEIAL APPENDIX. 623

therefore be disregarded, and'the quantity K will differ bnt
very little from unity also. From tliis it will be seen that
Wj -will differ the leas ft'om Q as Mj is greater than M,. But,
BB the mass of the cam shaft oi-dinanly very m^ich exceeds
that of the hammer, we can assume, without liability to any
great error, that the mean angular velocity of the cam ehaftj
deduced from observing the number of revolutions made by
it in a given time, is sensibly the ai'ithmetical mean of Q

and (jj. Designating this mean by ii, we have S2,= . — ^r— '.

From this relation and ec^uation (D) there obtains

„_2n,(^.+KM,). -, ^ ^ 2n,M,

SM^+KX,

and«,=^-

From these two relations the living force destroyed by the
impact can be deduced as follows. Before the impact the
living force of the cam shaft was H'M^B,"; after the impact,
as the point of contact of the cam and band moved with the
same velocity, the living force of the whole machine is

The living force destroyed therefore is expressed by

or, substituting for w^ from equation (D), by

o.-p.iM^M.'(M.+M,)L
■ I ■ (M,+EM.)' f -

™p, I (ag-l)M.M. + gM.- 1 .

• • 1 . {m,+e;m,)' t '

finally, substituting for S2 and w^ tieir values in Q,, there
obtains

2K-1+K'i
(aM.+KM.)- /,j.-kM.1

It is now readily seen, from the foi-m of this last expression
for the loss of living force by the impact, that, since K maj

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624 EDITOIMAI. APPENDIX.

be assumed as sensibly equal to unity, tlie numei-ical value

of this expt'ession will depend upon the ratio -j-^. Talcing

M,=:M, the value of the expression becomes lO^'MiEa'; *ud
for M,= o3 it becomes ^,"11,1^,', Tlieiefoie letween these
limits the difference is i only of the ]i\ing loice 1 st under
the Bupposition of M;,=oo ,

In the ordinary aiTangement of th s machine it laiely
occt,re that M, is not less than t>^M^. A sunnntf tins as the
limit, ajid substituting in the preceding expiets dr lOM lor
H,, there obtains for the required lose of hvmg force
0-9977si,'JI,Ej'. It is therefore seen that, in all usu^ caaee,
M, may be assumed as infinite without causing any notice-
ahle error in the resiilt.

To estimate the aceuniulated work expended bj the cam
shaft for each shock, n, &j, and n, being the same as in the
preceding expression, this work is expressed by

(ff-

As the earn shaft expends this amount of accumulated work
at each impact, a quantity of work equal to the half of tiik
must he yielded by the motive power at each impact, or

If therefore there are N cams on the shaft, and it makes n
revolutions in one minute, then the work consumed by the
numher of shocks in one second will be expressed by

"60 2M,+KM; ^>-

This then is the work consumed by the impact in one second
fo" the first period of the play of the machine ; and it has
been calculated according to what was laid down in Kote (t)
on the subject of shocks, by disregarding the work of the
other forces as inappreciable during the short interval of the
impact.

To estimate now the work expended dming the second
pei'iod, or whilst the cam and band are in contact after the
sU'^ck, let 0,G-, be any position of the line 0,0, during this

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EDITORIAL APPEXDIX.

period, making an angle G-iO,Gr=a witli its position when
tlie liammer is at rest. Represent "by P^ the nonnal pressure
at tlie surface of contact of the cam and band which will
balance all the resistances developed in the motion of the
hammer, leaving out of consideration that of inertia, as the
change of velocity between tlie end of the impact and when
the cam -disengages from the band ie so small that the living
force due to this interval may be neglected in comparison
with the work of the othei' forces ; by "W, the weight of the
hammer, its handle, &c.

When tlie line GO, is in the position GI-,C„ the line 0,if wiU
oe in that 0,^, making the angle tC^t,=:a. with its original
position. The force P, acting at t, in this position and per-
pendicular to the line i,C, — since the sortace of the band
produced pafises through the axis 0„ the surface of the cam
being an epicycloid— has for its vertical and horizontal com-
ponents P, COS. a and P, sin, a. The pressure on the trun-
nions of the hammer, which is the resultant of P, and W„
therefore will be expressed by

V'(W, + P, cos. ay + P," sin.' a. ;

and since the first tei-m of the radical is in all cases greater
than the second, the value of the radical itself may be-
expressed by (NotE! B)

7(W.+P, COS. a)+/3P, sin. B.

The equation of equilibrium between Pj and the other forces
will therefore be

P,Tl,="W,Gcoe. {a+a)+ jy(W,-l-P,cos.a)-t-,ep,sin.cc|^,8in-.i;>,.

Tlie moment of the friction at the point t^, due to P, with
respect to tlie point Cj, in tliia case from the form of the cam
and band, being 0.

As the pressure P, varies with the angle a, we can only
obtain its mean value by iirst finding its quantity of work
for the angle «=«! desciibed whilst the cam and band are in
contact. Multiplying the last equation by i^, and then
integi-ating between a=;0 and a=a, there obtaihs

( P,E,(^«=W,G{sin.(<H-c,)-sin.<(5 + \jW,(r.,+r'P^sm..ct,-'
'=0 liV^ COS. fi,+ 13^4 9, sin.. 9, ;

./Google

representing ty P^ the mean value of P, or tne constant
force applied vertically at t, which nmltiplied by R,a,, the
path described hj the point of application, will aire the
amount of work of tJie variable preaeure P, for tne same
path ; and introducing tliie mean value into the term of the
preceding equation that represents the moment of the fric-
tion on tne trunnions, as this will not produce any sensible
error in the results.

Now observing that the quantity G-jsin. («+a)— sin. a\ is
the vertical height through which the centi'e of gravity of
the hammei", &c. is raised during the period in question, and
that PmUjKi is the work of the mean force ; calhng this ver-
tical height A, and substituting the work of the mean for
that of the variable force in the last eqiiation ; there obtains

P™K,a,=W,A+{7W,«,+7P„sin.a,-/3P„cos.«,+
/3P„5 p, sin. (p,.

"W",A+yW,g,p, sin, ip.

~ R,a, — Sr sin. a, + /a (1 — cos.«i) \ p, sin. ^1 '

.(E).

If we now multiply the second member of equatioJi (E) by
It,ai we shall obtain the approximate value of the work of
the variable force P, during the period in question ; or the
value of PmRia, as detennined from equation (E).

To find now the work that the motive power must supply
to the cam shaft for tbis expenditm-e PmP,o:, due to the
motion given to the hammer during the period in question,
and also that arising from tiie resistances developed by the
motion of the cam shaft itself during this period, represent
by P, a force which, acting at a distance E, from the axis 0,
or the cam shaft, will balance 'all the resistances ai'ound C,;
by W, the weight of the cam shaft and its fixtures ; by 6 any
angle described by the cam shaft during the period con-
sidered ; and p the limiting angle of resistance at the point
of contact of the cam and band.

The pressure on the ti'imnions of the cam shai't is evidently
expre^ed by

■VT,+P,--P™;

and the equation that expresses the work of P, for the ele-
mentary angle dS js

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EDITOEIil. APPENDIX. 627

(W,+P,-P„)p, sin. <p, K^f^e.

Now representing by P. the mean value of P^, and substi-
tuting it for P, iii the last term of the second member of tliis
equation, which may be done without causing any sensible
eiTor in the result; observing, from the conditions of the
mechanism that Ej9=:It,rt, ; and integrating thk ecLuation .
between the limits d = 0 and 9=8i = — ^ ; there obtains, to
express the total work of P, for the angle 9„

=/p:b

(W,+P,-I'„) p, sin. i>, e,E,.

* Omitting tlie work oODsnmed bj the Mction of the axles in equation (SSI)
(Art. 230), that which ia expended on the teeth, in contact whiUt tie aro r,^ ii
described is represented by the term of the equation

Dividing this lost exprei

ae the value of a mean or constant force which applied langentially to the
cironinference haviog the radiua r^ will expend, whilst the point of applicollon
describes the are rjijj, the same quanljty of work as that consumed hy the fric-
tion of the teeth in contact whilst this arc is described. In this expression
the value of Fg is less than the true value.

The foregoing is the theorem of M. Poncelet referred to on page lii.
Author's Preface. The direct manner of deducing it is fonnd on piige inS
Navier. Resume des Lei;ims^ Ac. Troisi^me Partie. Paris, 1833.

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EDITOKIAL AFPRsmS.

Kl\ '2 ' ■ - - - " . . . (F)

The -work therefore that the motive power must supply to
the cam shaft during this period is found by multiplying tho

second member of eq^uation (P) by B,9,=:Rj-^ or the path

passed over by the point of application of the mean force P,
during this period.

Representing in like manner, by -^ the number of times

the hammer is raised per second, the quantity of work tliat
the motive power must supply for this expenditure will be
expressed by

60 " " 60 ' ' K, ^ ^

liuring the last period, or whilst the hammer ia down, the
motive power will only have to supply the expenditure of
work caused by the friction on the trannions of the cam
shaft, arising from the weight of this shaft and its fixtures
and tlie power; any accumulation of work in this shaft
during this period being neglected as small in amount.
Representing by j>=K flie number of cams on the shaft, their
distance apart on the primitive circumference whose radiua

is R, is evidently -> and, as the arc described on this

circumference whilst the cam shaft and hammer are engaged

3^R,

is K,ff,, that described wliilst the hammer is down is

. . . . P

E,ce,. Calling Pp the power which acting at the distance R,
will balance the friction arising from the weight W, of the
cam shaft and fixtures and F„ me value of Pj, will be found
according to the conditions stated as follows,

Pj,-K,=(W,+Pp)p,sin.9,.

TheworkofPpii

^R=-

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EDrmRIAL APPENDIX.

as the path passed over by its point of application is evi-

dently the are ^ I ±t,a, I .

The work which the motive power miist supply therefore
per second during tliis last period is expreBsed by

By taking the sum of the quantities expreesed by the
formulae (I), (2), and (3) there obtains

3M,+KM, ' - = ' ' ^K^\

s the total work that the motive power must yield
to the cam shaft per second to supply the work consumed
by all tho resistances.

Hat consumed by the useful resistances, which consist of
half the living force ti'ansmitted to the hammer and the
work consumed in raising the centre of gravity of the ham-
mer, &c., through the vertical height h is represented by

~2 + ™ '* -(2M,+KJI3-+ '*•

From the preceding expressions, it is easy to deduce the
wort which must be expended in producing a given depth
of indentation by the hainmer upon the niet^ when brought
to a given state of heat. For this purpose, we observe that
to h^ the living force acquired by the hammer there cor-
responds a certam amount of work, estimated in terms of
the weight of the hammer and a certain height A, to which
ite centre of gravity hae been raised, and expressed by

the total work therefore expended by the hammer in
indenting the metal is expressed by W,A,+ W,A ; since, from
the state of the metal the molecules whitii are displaced by
the impact acquire velocities which are not appreciable from
their smallness; the resistances therefore offered by the
metal to indentation may be regarded as independent of the

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630 EDrroEiAL appebdix.

velocity and, from the laws of the penetration of solids intc
different media, proportional simply to the area of the inden-
tation. Representing tbeo hy « and 5 the eidee of t!ie area
of Uie indentation, supposed rectangtiiar, at the enrface of
the metal impinged on, d the depth of the indentation, and
C the constant ratio of the resiBtance and the area of the
indentation, the following relation obtains bet-ween the work
expended by the hammer in its fall and that offered by the
resistance of the metal

an equation from which 0 may he detei-mined hy experi-
ment in any particular case.

It will be readily seen that the preceding expressions will
he rendered applicable to the cases where the cam catches
the hammer on the same side of its axis of rotation as its

centre of gravity, by writing — -^J- MG for + -^MG, and
dz dt

moreover in this case when P— -^ MG=0, there will be no

dt
shock on the trunnions (Arts. 108, 109), and there then
obtains, to find the point where the cam should catch the
hammer con-esponding to this case,

• Morin, Suite des Noaeelles Ext

, Google

APPENDIX.

Theoehm. — The d^iie integral j fxdx ia ih« limt of the turns 0/ iJu

vahies secerally aasumed 'by tke prodw;t fx , Aa, as x is made to vary hy
tueceisive equal increments of A», from a to 8, a/nd os eocA such equal
increTomit U continually a/nd infinitely ditnmished, and their nurtiber there-
fore eontinudlly and infinitely in/ireaied.

To prove this, let the general integral be represented by Fs; let us sup-
pose that/B does not become infinite for any value of x between a and 6,
and let any two sueh values be x and x + Ax\ therefore, by Taylor's the-
orem, r (iB + dx) = Fs + Axfe + (ii!) ' ■•■ XM, where the exponent 1 + ii is
^vento the third term of the expansion instead of the esponent 2, that the

case m;^ be inclnded in which the second differential coefBdent of Pic, -^,

is infinite, and in which the exponent of Ax in that term ia therefore a
fraction less than 3.

Let the differeEce between !t and i bo divided into n. equal parts; and
let each be represented by ic, so that

Giving to w, then, the successiye values a, a+ ai;, a + 3is . . a + (n — 1)
Ai, and adding,

F(o + JiAa)=Fa + AiB2,y{a + (7i— l)aic} + (iic)'-i-)>2it.,

.-. n—Sa=Ax^,'f{a^{n—\)Ax} +{Axf*\-Lii,.

"Sow none of the values of M are infinite, since for none of these values is

^infinite. If, th6refore,Mbethegreatestof these values, then is sM,less

than nUL: and therefore

F6— Fa— A«X,'j^a+(ji— l)Aai><(S— «)M(Aa)x.
The difference of the definite int^al F6 — Fa, and the snm S,"(is)/{«-f-
(«— 1) Ax) is always, therefore, less than (6— a) JI(As)». Now M is finite,
and (5 — a) is ginen, and as m is inoreaaed A« is diminished continually ;
and therefore (aa!)X is diminished continually, % being positive.

Thus by increasing n indefinitely, the difference of the defloite integral

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633 Al'PENDIX.

and tie sum may 1)6 diminished indefinitely, and tberefora, in tho limit, tha
definite integral is equal to the Bum (i. c.)

FS— Pa = Umit s.,"{Ax).f{a+(n~l)Ax};
or, interpreting this formula, Yi — Fa is the sum of the Talnes of A* .fc,
when « 13 made to pflas by infloitesimal increments, eacii represented by
ii, from a to 6.

KOTE E.
Pohoblbt's Fiest Theoekm,

* The Takes of a and & in the radical ■•/a^ + b' being linear and rational)
let it bo reqnired to determine tho values of two iadetemiinate quantities
a and3,sucli that the errors which result from asBumingy'oi'+fi'=a«+/35,
tlirongh a^ven range of the values of tlie ratio! ^j, may be the least pos-
sible in reference to the trae value of the I'adical ; or that ° ^ —

Va'+S'

—1, may be the least posdble in respect to all that range of
values which tbis formula may be made to assame between two givea
estreme values of the ratio i. Let these esti'eme values of the ratio j
be Tepresented by cot. i^i and cot. i^ and any other value by cot. ^. Sub-
stituting cot. 4 for 1 in the preceding formula, and observing tha,t-\/ a'-^-b'

=-}/h'aoi.''i+f>'=h cosec, 4, also that aa+p6 = a3 cot. ++|}S=(b cos. ^H-B
sin. 4)i cosec. ■i, the corresponding error is represented by

ttC03.4+)j9in.4— 1 (1);

which expression is evidently a maximum for that value i^s of tp wbich is
determined by the equation

<Mt.>p,=^ (2);

V«."+3'— 1 (S).

Moreover, the fanction admits of no other maximum value, nor of any
minimum value. The values of a and (3 being ai-bitrary, let them be

assumed to be snob tliat^or cot. if'j may be less than cot. t^,, and gi'oator

* The method of tbis iuvestigation is not tlie same as that adopted by IVL
Ponoelet ; the principle is the same.

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POKCELiri's THEOEEJI. 633

than cot 4/,. "Sow, 80 long as all the ralues of the error (forranla 1)
remain positire, between the proposed limits, they we all manifestly di-
minished by dimiiiisting o and (3 ; hnt when by this diminution the error
is at length rendered negative in respect to one or both of the extreme
values 4„ or 4, of 4, and to others adjacent to them, then do these nega-
tive errors oontinually inorease, aa a and )J we yet farther diminished,
whilst the positive maximum error (formula 3) continually diirtiniihea.
Now the most favorable condition, in respect to the whole range of tha
errors between the proposed limits of variation, will manifestly be attained
when, by thns diminishing the positive and thereby increasing the negative
errors, the gi'eatest positive error is rendered eqioal to each of the two
negative errors ; a condition which will be found to be c!)ll^iste^lt with
that before made in respect to the arbitrary values of a and /3, and which
sapposes that the Viilues of the en'or (formula 1) corresponding to the
values +1 and +j are each eqnal, when taken negatively, to the maximum
error represented by formula 8, or that the constants a and S are taken
BO as to satisfy the two following equations.

l-(a COS. V,+^ sin, ■^,)~ V^+f'-l.
1-fy, 0O3. *,+p sin. *,)=1— (f COS. -*-j+(3 sin. ■*■,).
The last equation ^ves us by reduction

s. -tfi + lSsin. ■4',=i3

anda = 3 cot. ■i('*'i + *J.
Snbstituting tiese values in the first equation, and reducing,
asin. K^,+f.) _ain.i(-ir, + ^,)

'^~l-(-003.i(*,+ -9^ cos.'i(*, — ■4',) ■■ ■''

. _ 2oos.^(^,+-g,) _cos.K^,— ^J

* 1+cos. ■i(*, — *-^ oos.'^(*,— *,) '• -''

These values of a and 0 give for the maximum error (formula 8) the ea-
pi'ession

tOE. >«*,-■!',) (6).

Thus, then, it appears tliat the value of the radical v'a' + S' is represented,
in respect to all those values of t which are included between the limits
cot. 'l"] and cot. ^F,, by the formula

„cos.i(^,+^.) ^n.K-^,+-F,)

with a degree of approsimation which is determined by the value of

If in the proposed radical the value of a admits of being increased in-
finitely in respect to 5, or t3ie value of J infinitely diminished in respect to
o, tJien cot. t, = infinity ; therefore *, = 0. In this case the formala of
approximation becomes

, Google

634

a(l— taTi.H'^'i) + 25tan.iT, (8);

and the masinmm eiror

tan.=i*, (9).

If the values of a and 5 are wholly unlimited, so that a may bo infinitelj
Hmall or infinitely great as compared with S, then oot. 'Fi = infinity,

0;

the formula of approx-

■ (10);

cot. -i'i :

therefore ■4'i=0, ^Fs=o- Snbstitntiug these
ijnatioa becomes

■8284a+-8284&

and the niaximiim error

■171B, or ^th nearly.

If h is essentially less than «, bat may be of an^ value less thaQ it, s

that T is always greater tJmn unity, but may be infinite, then cot. if-, ^ ir

finity, cot. ifi=l ; therefore i^i=0, V'i=7. Substituting these values in th
formula of approsimation, and reducing, it becomes
■960i6ii+-39783S (H);

and tlie

■039-15, c

jVth n

It is in its application to this case that the foramla has been employed ii
the preceding pages of this work.

The following table, calculated by M, Gosaelin, contains the values o
the ooefEcienta a and 3 for a series of values of the inferior limit oot. <p„ th
superior limit being in eveiy ease injinity.

HelatlonofotaS.

If

^5

Talneofj.

VBlnnoffl.

.„,„„

Error.

of v^ni

a and b any )
whatever f

0

0-82840

0-82840

0-17160

■•J

0-8284 {a + 6)

«>i

1

0-98046

0-39783

0-03954

1-.V

-960460 +

897836

o>26

2

0-9a692

0-28270

0-01408

■98692a +

232706

a>%h

3

0'9338O

0-16X2S

0-00650

^-h

■993500 +

161236

a>ib

4

0-99625

0-12260

0-0037S

■•4^

-996260 +

122606

B> 66

5

0-997 B7

0-09S78

0-00243

-997B7a +

098786

a> 66

6

0-99828

0-03261

0-00174

^^u

■99S26a +

082616

a>nb

1

099875

0-07098

000125

'^h

-99S7Ba + -070986

a>Sb

8

0-99905

0-08220

0-00095

^Wt^

■999060 + -062204

«>96

9

0-99030

0-0658S

0-00070

'tA.

■999300 + -0663Bi

a> 106

10

0-99986

0-04984

0-00065

■■t^'..

■999S6n + -049846

, Google

POMCELET S SECOND THEOEEM,

Pohoklet's Sboonb Theobsm,

To approximato to the value of fa"— 5", let iwi — Si be the formula o(
approsimatioc, then will the relative error he represented by

Now, let it he observed that a' being essentially greater than 5', ~> 1 ;

let J. therefore, te repreaonted hy coseo. 4, then ivill tie reliitlve error ba

represented by 1— -^— :zz=^=-, or by

1-aseo.^ + etan,.}. (12),

which functiott attains its masiraum irheu sin, 4, = 2 Substituting this
value in the preceding formula, and observing that —a. sso 4 -j- p tan. 4. =

— 3ec4(a— J3sin.4)=~— ;^=^=— f'a'_3^, we obtain fo. the maximum

eri'Or the espresrfon

1--V^=^' (13),

Assuming ^ and 4i to represent the values of 4, correspociUfc^ to the
greatest and least values of t-, and observing that m this case, as in tbe

preceding, the values of a and j3, which satisfy the condition? of the
question, are those which I'ender the values of the error corresponding to
these hmits equal, when t^ken with contrary signs, to the maximum error,

— 1 + 0 sec. +1— 3 taa. 4, = 1 — Va'—S' (14).

1 ~a sec. 4-14-0 tan. 4:,=l—aBeo.4j + S tan 4, . . . . (ID).
The latter equation gives, hy reduotJon,
COS. i(4,— 4,)

i.*(4'. + 4.)"'

■ 'Mj

in.*i(^i + *,) n •" rin.'KV'.+*,)•
Anda8ec. !f,-[3tan. i/y,=3cot.i-(tj'i+i/,5) .... (IT),
Substituting these values in equation (14), and solving in respect to fl

./Google

^^.^yr}.;^.'^" ■■ .... (18).

o^.J(lf'| + l^,)+ Vcos, i/'iCoa. ij-j
•or 13 represented by the formula

. (19).

COS. i {ipi + 1/',) + y COS. ii-, COS. 1^,

These foiTnuIic will be adapted to logaiithmic calculation, if we aasume

i (1J-1 + !j,J='P|. and ^."^r.^ ytl~ '^'} = eoaec. 'f,; we shall thus obtain from
sm. i(i^| + i^j)

equations (16) and (17)a = 0 coseo. ^j, Va' — /3' = 3 cot, 'f „ and a sec. i;.;
— 3 tan. ^^, = d oot. *i ; therefore, by equation (14),
2 Sain.-'I'iain*',"!

t. U', + cot. %.^ sin. (% + *,) I
t. >f , + cot. 1's ~ sin. (f , + '!',) J

. . (21).

Maximum ei

^- (,%-%)

sin. (*, + yj (22).

The form under wliicb. this theorem has been given by M. Poncelet ia
different from tke above. Assuming, as in the previous case, the limiting

values of ? to be represented by cot. i/-, and cot. 'P,, and proceeding by a
geometrical method of investigation, Le has shoten that if we assume

2CPS-YI 2cos.'y, . _ sin, (yi — rJ

"-sin.Cv.+y,)''' sin. (y,+y,) cos.fi' ail.(y, +y^*

If the least possible value of a be 1-^^h, and its greatest possible value
be infinite as compared Trith S, M. Poncelet lias shown the formula of
approsimatiou to become

Va? — &' = l'1819(i— 0-Y2686S (23),

with a possible error of 01319 or ^ nearly.

If the least possible value of a be 2S, and its greatest possible valna
infinite compared with h ■ then

V^^^;T' = 1-018628(i — 0-3T29M6 (24),

witli a possible error of -0186 or ^'^d nearly.

./Google

ON THE ROLLIXG OF SHIPS.

(First published 6y tfte Autltor in the Trmnsactlom of the Soyal Society

for 1850, Fart II.)
Let a body be conceived to float, acted upon by no other forces than its
weiglit W, and the upward pressure of the water (equal to its weight);
which forces may be conceived to be applied respectivelj to the ceuti'e of
gravity of the body and to tlie centre of gravity of the displaced fluid;
and let it be supposed to be subjected to the aoWon of a tliird force whose
direction is parallel to the surface of the fluid. Let aH, represent the ver-
fioal displacement of the centre of gravity of the body thereby produced*,
and iH, thfit of the centre of gravity of ita immeraed part. Let more-
over the volume of the immersed part be conceived to remain unaltered t
whilst the body is in the act of displacement. If each centre of gravity
be assumed to ascend, the work of the weight of the body will be repre-
sentad by — W.iHi, and that of the upward pressure of the fluid by +
W,aH„ the negative sign being taken in the former cose because the force
acta in a direction opposite to that in which the point of application is
moved, and the positive sign iu the latter, because it acts in the same direc-
tion, ao that the aggregate work SMj (see equation 1, p, 122.) of the forcea
which constituted the eqnilibrimn of the body in the state from which it
has been disturbed is represented by

— W.aH,+T.aHj.t
Moreover, the system put in moHon includes, with the floating body, the
particles of tlie fluid displaced by it as it changes its position, so that if
the weight of any element of the floating body be represented by w„ and
of tlie fluid by m„ and if their velocities be », and v.„ the whole vis 'dtia is
represented by

• "When a floating body ia ao made to incline from any ods position into any

the one poaition be
Lty ia also vertically displaced ;
of the oi

other as that the voluma of fluid displaced by it

equal to that in the oHiei

for if this be not the ease, the perpendicular dist

of the body from its plane of flotation must remein nniihaiiged, and the form

of that portion of itaancfnee, -whioliiB snbjeottoimmeraion, must betieiiDnined

geomOrieally by this condition ; but by the suppoaitiou the form of the body

ia undetermined. It is remarkable what currency bus been given to the error,

that whilst a vessel is rolling or pitching, its centre of gravity remains at rest

I should not otherwise have thoi^ht this note neoeeaaiy.

•[• TTiiB auppoaitiou is only apprratiraately true.

% If the centre of gravity of the body or of the displaced fluid tkscends (t
property which will be found to oharaoteriBB a lai^e class of vessels), AH, iu
the one case, and aH, in the other, will of coarse take the negative sign.

, Google

e have by equation 1 (p. 132),
U(e) — -w(ah:, — aHO = wzZ

sr

In the estrarae position into wMoii tlie body is made to roll and ic

Tim^w/iin,— Ano+i^2w,i!L (26).

ot if tt© inertia of tie displaoaci fluid te neglected,

U(s)=W.(aH, — aH,) (37).

Whenee it follows that the vioi-Js neessioiry to incUne a floating iody
thrmigh om^ gwea. angle is eq'oal to that necessary to raise it Itodih/ throitgh
a height egwtl to the d^ffh'enee of the verUeal displacements of its centre
of gi-avity and of that of its imanersed part ; so ^Mt other things ieing
the same, (Aa( ship is the most stable the product of vihose weight by Giis
difference is the greatest.

In the ease in ■whioli the centre of gravity of the diaplaoed fluid deaoends,
the sum. of the displacements is to be taken instead of the difference.
This conduaion is nevertheless in error in the following respects: —
lat. It snppoaaa that throughout the motion the weight of the displaced
fluid remains equal to that of the floating body, which equality cannot
accurately have been preserved by reason of the inertia of the body and
of the displaced flnid,*

From this cause there cannot but result smaU vertical oscillations of the
body about those positions which, whilst it ia in the act of juclming, cor-
reapond to this equality, which oacillntiona are independent of its piincipal
oacillafaon.

2ndly. It involves the hypothesis of absolnte ri^dity in the floating
body, so that the motion of every fart and its sis vka may cease at onee
when the principal oscillation terminates. The frame of a ship and ita
maats are, however, elastic, and by reason of this elasticity there cannot

* The motion of the centre of gmvity of the body being the same as though
all the disturbing forces were applied directly to it, it follows, that no elevation
of this point is oavised in the beginning of the motion, by the application of a
horizontal disturbing force, or by a horizontal displacement of the weight of
the body, which, if it be a ship, may be effected by moving its bnllKst. The
motion of rotation thereby produced takes place therefore, in the feat instance,
about the centre of gi-avity, but it cannot bo take place without destroying the
equality of the weight of the displaced fluid to that of the body. From this
inequality there results a vertical motion of the centre of gravity, and anothel
aJtis of rotation.

, Google

ON IlIL KOIIING nr sinps. 639

but result oscilltitions, ■which are indepenilent of, ftnd mny not synchro-
nise with, the principal oscillation of tlie sliip as she rolls, so that the i>k
«W(t of every part cannot be a'isumpd to cease and detennine at one and
the same instant, as it has been supposed to do.

Srdly. Ko account has been taken of the work expended in communi-
cating motion, to the displaced fluid, measured by half its im viva and

represented by the term ^SMjuJ in equation 26.

Trom a careful consideration of these causes of error, the author was
led to conclude that they would not affect that practical application of the
formula which he had principally in view in iuTestigating it, especially as
in certain respects tliey tended to neutralise one another. The question
appeared, however, of sufSciant importance to be subjected to the test of
esperiment, and on his application, the Lords Commissioners of the Admi-
ralty were pleased to direct that such esperiments should be made in Her
Majesty's Dockyard at Portsmouth, and Mr. Pinoham, the eminent Master
Shipwright of that dockyard, and Mr, EiwSON, were kind enough to
undertake them.

These experiments extended beyond the object originally contemplated
by him ; and they claim to rank as authentic and important contributions
to the science of naval consfmction, whether regard be had to the prac-
tical importance of the question under discussion, the care and labor
bestowed upon them, or the many expedients by which these gentlemen
succeeded in giving to tliem an accuracy hitherto unknown iu experiments
of this kind.

That it might be determined experimentally whether the work which
must be done upon a floating body to incline it through a given angle be
that represented by equation 27, it was necessary to do upon such a body
an amount of work which could be measured ; and it was ftirther neces-
sary to asoertwu what were the elevatlous of the centres of gravity of the
body and of its immersed part thus produced, and then to see whether
the amount of work done upon the body equalled the difference of these
eleTations multiplied by its weight.

To effect this, the author proposed that a vessel slioidd be oonsfrncf«d
of a ample geometrical form, such ttiat the place of the centre of gravity
of its immersed part might readily 1>e determined in every position into
which it might be inclined, that of its plane of flotation being supposed to
be known ; aud that a mast should be fixed to it and a long yard to this
mast, and that when the body floated inavtalp aw ght

suspended from one extremity of the yard should sudd nly be allowed to
act upon it causingit to roll over; that the po ti n nto h h t thns
rolled should Ije ascertained, together with the po d ng 1 at n

of its centre of gravity and the centre of grav ty f ts mm rs d pa t
and the vertical descent of the weight suspend dtmth tmjf
its arm. The product of this vertical descent by tl w ht uspended

, Google

640 APPENDIX.

from tho arm ought then, by the formula, to be found nearly equal to ttie
difference of tie elevntions of the two centres of gravity mnltiplied by
the weight of the body ; and this was the test to which it was proposed
that ike formula should be eulgeoted, with a view to its adoption by prac-
tical men as a principle of naval construction.

To give to the deflecting weight that ivatantaneovs action on the ex-
tremity of the arm which was necessary to the accuracy of the experiment,
a ati-lng was in the first place to be afflsed to it and attached to a, point
verlJoally above, in the ceiling. Whan the deflecting weight was first
applied this string would sustain its pressure, but tliis might he thrown
at once upon the eatremity of tlie arm by cutting it. A transverse seo
lion of the vessel, with ita mast and arm, was to he plotted on a large
scale on a board, and the esfreme position into whioli the vessel roEed
being by some means observed, the water-line corresponding to this
position was to be drawn. The position of the yard, in respect to the
surface of the water in that position, would then be knoivu, and the vertical
descent of the deflecting weight could be measured, and also the vertical
ascent of the centre of gmvity of the immei-sed part or di^plnoement.

To determine the position of the centre of gravity of the veaael, it was
to be allowed to rest in an inclined position nnder the action of the deflect-
ing weight; and the water-line corresponding to this position iieing drawn
on the board, the corresponding position of the deflecting weight and of
the centre of gravity of the immersion were thence to be detenuined.
The determination of the position of the vertical passing through the
centre of gi'avity of the body would thus become on elementary question
of statics ; and the intersection of this line, with that about which the
section was symmetrical, would mark the position of the centre of gravity.
This determination might he verified by a second similar experiment with
a different deflecting wiaght.

These suggestions received a great development at the bauds of Mr.
EiwsoN, and he adopted many new and ingenious espedienta in carrying
them out. Among these, that by which the position of the water-line
was determined in the exti'eme position into which the vessel rolls, ia
specially worthy of observation. A strip of wood was fastened at right
angles to that extremity of the yard to which the deflecting weight was
attached, of sufficient length to 3ip into the water when the vessel rolled ;
on this slip of wood, and sdao on the side of the vessel nearest to it, a
strip of glazed paper was fised. The highest points at which these strips
of paper were wetted in the rolling of the vessel, were obviously points
in ttie water-line in ita extreme position, and being plotted upon the board,
a line drawn through them determined that position with a degree of
accuracy which left nothing to be desired.

Two forma of vessels were used ; one of them had a triangular and the
other a semicircular section. The following table contains the general
results of the espeiiraonts.

, Google

ON THE EOLLISGr OF 8IirF5.

ss

■s'

S,'?Li'

St

1

1
1

s

i

i

11

1
ill

ill

m

»«■■

1.
8.

31-3568
a82Bll

i

i«n(

■ssei

as 30

I]

■mo

modeL

i.

ill

VWlt

i-7Tai

7-SM

is

K

1i

In the experiments witli the Braaller triangular model tte differences
between the results and thoM given by the formula are ranch greater than
in the experiments witi tlie heavier cylindrical vessel.

In espknation of this difference, it will be observed, ^raf, that the con-
ditions of the experiment with the cylindrical model more nearly approach
to those whioh are assumed in the formnla tlian these with the other; the
disturbance of the water in the change ot the position of the former being
less, and therefore the work espended upon the mertia of the water, of
which the formnla takes no account, less in the one car* than the other;
and, &m(mdly^ that the weight of the model being greater tins inertia
bears a less proportion to the amount ol work re ]uired lor inclining it
iiian in the other case.

The effect of this inertia adding itself to the buoyanoT of the fluid,
cannot but be to lift the vessel out of the water and to use th d splace-
ment to be less at the termination of each rolli g os illati n tha at its
commenoement.* This variation in volume of th d pla m t w s appa-
rent in all the experiments. Its amount was m asn d a 1 corded
in the last column of the Table ; its tendency i to j d in the body
vertical oscillations, which are so far indepecd nt f t II a notion
that they will not probably synchronise with it. The body,, displacing,
when rolling, less fluid than it would at rest, the effect of the weight
used in the esperimenfs to inohne It is thereby increased, and thus is
explained the &ot (apparent in the eighth and ninth columns of the Table)
that the incUnation by esperiment is somewhat greater than the formula
would make it.

The d/ynamiioal sti&iliiy of a vessel whose athwart seetiona (ph^e they

* This result connects itself with the well-known fact of the rise of a Teasel
ont of the water when propelled rapidly, whicli is so great in the case of fust
troo^-boats, as considerably to reduce the resistance upon them.

, Google

642

are sm5j^( to immm'don and emersion) mre eir&ular, luizing their centres w

iwliose secti

Let EDF, fig. 1. or 3., be an athwart section of such a vosael, &e
parta of whose periphery ES and TB, subject to immeraion and emersion,
are parts of the same wronlar arc ETF, whose centre is 0. Let G, repre-
sent the projection of the centre of gravity of the vessel on this section,
and G, tliat of the centre of grayity of the space whose section is'SDET,
supposing it filled ■with water. The space lies wholly within the vessel in
fig. 1. and withont it in fig, 3, Let

Ai = CG„ A, = CG,.

W, = weight of vessel.

eight of water occupying, or which would occupy, the space
IS STBD.

B = the inclination from the vertical.

Since in the act of the inclination of the vessel the whole volnme of
the displaced flnid remains constant, and also that volnme of which STED
is the section,* it ibilows that the volnme of that portion of which the
oironlar area F8BQ is the secdon remains also constant, and that the
water-line PQ, which is the chord of that area, remains at the same dis-
'.tanoe from 0, so that the point C neither ascends nor descends. "Saw the
forces which conatitnted the equilibiiam of the yasael in its vertical posi-
,tion were its weight and that of the fluid it displaced. Since the point 0
is not vertically displaced, the work of the former force, as the body
inclines through the angle 9, is represented by — "W, h, vers. e. The work
of the latter is equal to that of the upward pressure of the wal«r which
■would occnpy the space of which the circular area PTQ is the section
^iiioreased, in the case represented in flg. 1., by that of the water which
would occupy STED ; and diminished by it in the case represented in

:ae. 3.

Bnt since the space, of which the circnhr area PTQ 3 the eot on
remains similar and equal to itself, its ce tre of grav ty reu a ns alwava
at the same distance from the centre C an 1 tb refore ne ther as ends
nor descends. Whence it follows that the work ot the wile ■nl cl
would occupy this space is zero; so that the w rk of the lole d s] laced
iuid is equal to that of the pa/rt of it wh h occnp es the space '^TED

* It win be obsevyed that the apace STRD j; j. od alw ys to be undet

, Google

OF SHITS. (!i3

taken in the case repreBeiitefl in fig, 1, with the poaUve, and in tliat ve-
prasented in fig. 3. with tlie negative sign. It is represented ttei-efore,
genei'ally by tlie formula ±'W,hs\eis.S. On the whole, therefore, the
work SM, of those foi-ees, which in the Tertioal position of the body con-
stituted its equilibrium, is represented by the formala —
2'Ms = — W, A, Ters.e ± WjAjVers. fl.
Representing, therefore, the dynamical atahihfy 2«i by U (9), we have by
equation (2. p. 122.)

U (9) - C^. *i T 'W, ft,) vers. 9,
in which expression the Bign '-F is to be taken according as the circular
area ATB lies wholly within the area ADB, as in fig. 1 , or partially with-
out it, aa in fig. 3. Other things being die same, the latter is ibHrofore a
more stable form than the other.

18. The work of the upward pressure of the water upon the vessel
represented in fig. 3. being a negative quantity, — W, ftj vers. S, it follows
that the point of appUoation of the pressure must be moved in a direction
opposite to that in which the pressure acts ; but the pressure acts upwards,
therefore its point of application, i. e. the centi'e of gravity of the displaced
fluid, descends. This property may be considered to distinguish mecha/ni-
ccUly the class of vessels whose type ia fig. 1., from that class whose type is
fig. 2. ; as the property of including wholly or only partly, within the area
of any of tiieir athwart aeotJons, the corresponding circular area ETF, dis-
tinguishes them georaetiically.

7%e dynamicalstaMlity<if a vessel of any gwen form sheeted to a roll-
mg or pitching motion.

Oonoeive the vessel, after having completed an oscillation in any ^ven
direction — being then about to return towards its vertical position — to
be for an instant at rest, and let E3 represent the
interaeotion of its plane of flotation then, and PQ
of its flotation when in its vertical position, with
a section OAD of the vessel perpendicular to the
mutual intersection 0 of these planes. The sec-
tion OAD will then be a vertical section of the
vessel.

Let 6 he the projection upon it of the vessol's
centre of gravity when in its vertical position,

H, that of tlie centre of gravity of the fluid displaced by the vessel in the
vertical position.

g, that of the fluid diaplaced by the portion of the vessel of which QOS
is a section.

k, that of the fluid which would be displaced by the portion, of which
POE is a section, if it wei'e immersed.

GM, HW, jm, A», KL, perpendicnlars upon the plane R8.

■W=: weight of vessel or of displaced fluid.

w = weight of water displaced by either of the equal portions of the
vessel of whioh POK and QOS are sections.

./Google

Hi = depth of centre of gi'avity of vessel in vertiOHl position.

Ill = depth, of centre of gravity of displaced water in vei'tica!

aH| = elevafjon of centre of gravity of vessel.
aHi = elevation of centre of gravity of displnced water,
P = area of plane PQ.
fl = iaolinatjon of planes PQ and E8.
ij =: inclination of line O in wliich planes PQ and ES intersect,

to that line about whicli the plane PQ is sjminetiicjil.
h = perpendicular distance of line O from centre of gravity of

plane PQ.'
f = incUnafdon to torizon of lino about wbieb tbo plane PQ is

symmetrical.
<e = distance of aeotion CAD, measured along the line ■whose
projection ia 0, ftom the point where that line inferseols
the miilship section.
!/ = 0i3.
J/. = PQ.

2 = ftm + mg.

). = KL.

I = moment of inertia of plane PQ about axis 0,
A and B = moments of inertia of PQ abont its piincipal axes.

ft = weight of a cubic tinit of water.
Suppose the ■water actually displaced by the vessel to be, on the contrary
eontaimd by it; and conceive tliat which occupies the space QOS to pas&
into the space POP, the whole becoming solid. Let iHj represent tlie
corresponding elevation of the centre of gravity of the whole contained
flnid. Then will iHj + iiHj repre.'^ent the total elevation of the centre of
gravity of this fluid aa it passes from the position it occupied when the
vessel was vertical into tte position PAQ. Bnt this elevation is obviously
the same as though the fluid had assumed the solid atat« in the vertical
position of the body, and the latter had revolved ■witli it, in that state, into
its present position. It is therefore represented by EH — Nil ;

.-. aH, + iH, = KH — ¥H and aH, = KH — :SH — aH,.

r, by the elevation of the fluid in QOS, whose weight is m,
into the apace OPE, and of its centre of gravity through (ffm + Aw), the
centre of gravity of mass of fluid of which it forms a part, and whose weight
ia "W, ia raised through the space AH, ; it follows, by a well-fcnovra property
of the centre of gravity of a system,* that

* Tlie line joining the centres of gravity of the vessel and its immersed part,
in its yertioal position, is parallel to the plane CAD, for it is perpendiuular tft
the pkne PQ, to whose intersection with tbe plane ES the plane CAD is per
pendicHlar; , ■. 6K = Hi and HE = II,.

./Google

ON THE ROLLING OF SHIPS.

.-. KH— KH = II, vera. S +}.,

.■.W(H,yers.e+j.-An,) = WJ;

.■.W.AlIs = W(H,Tera.a+).)— 8*2 (28).

Also aH, = KG— MG = H, — (H,co3.9— A) = H,yers.S+^;

.■.W(iH,-iH,) = W(H,— H,)vers.o+«i2;

.■.(eqaationar.)TJ(e,.?) = W(H,— H5)vsr3.a+iw; . . . (29).
If a)J be a vertical prismatio element of the space QOS, whose bas
is &t ^ COS. fl, and height y sin. 6 thea ■will id. mg be reproBunted, i

reapeot to tbat element, by ny sin. B. dee dy oos. fl. ^ y si
.anted, in respect

s. ^fyfdjx dy,

COS. e y'dai dy ; and viz will be represented, in respect to the ■whole space
of whioh PrsQ is tlie section, by

If therefore we represent by $ the value of icz, in respect to the spaces
of which the mixtilinear Jiraas FBs' and QSs ai-e the sections, we have

MZ=:ni"'Isin.'9C0S.9 + 4j.

Bat the axis 0, about which the moment of inertia of the plane PQ ia

I, is inclined to the principal axes of that plane at the angles ij andn — ij,

about which priacipal axes the moments of inertia are A and B,
.■.I = Acos.'^+Bain.=^+PS,',
.-.U(e,,)=

"W(H,-H^vers. e + 2>(Aoos.';, + Bsin.',+PA')siii'9oos.fl + f ...(80),

It has been shown by M. Dbpib* that when 6 is small the line in

* Suria Stability des Corps Flotfants, p. 32. In oalotilatJona having refe>
enoe to the stability of ships, it is not allowable to oonsiiJer 0 extremely small,
except in ao for as they have reference to the form of the ship immediately
about the loadrinater line. The rolling of the ship often extends to 20° or 80°,
and is therefore largely inflnenoed by the form of tlie vesael beyond these
limits. Generally, therefore, equation 80. ia to be taken as that applicable to
the TOlUng of ahipa, those which follow being approniiaationa only applicable
to small osdllatiotis, and not eufficiently near (eseepting equation 37) fol
practical purposes-

, Google

■which the planes PQ or ES intersect passes through, the centra of graiitj
of each ; in this eass

.■.I = Aco3.'^ + B3iiL,= ,;
therefore by equation (80),

UCfl,il) = W(H,-HJvers.a + 2f'(Acos.', + Bsin.'^)sin.'tfcos.fl+4,.

If fl be 80 small that the spaces Pi'R and. Q^ are eyaaesoentin compaji-
soa with POr and QOs, then, assuming ^ =: 0 and, cob. fl ^ 1,

TI(«,,) = W(H,— HjTei..»+-2>(Aco..',+Boii.-,).m.'ii,...(31),

■which may be put under the form

r(e,^)= -jWip, — H,) + ^(Acos,'>, + Bsin.'^) l vors.fl.

sin. r = Bin. 9 sin. ^,,...(82),
and (Aoo3.= , + B3in.'^)ain.'e = {A + (B — A)sin."^}sin.'S,

.■.CAcos,', + Bsin>,))sin.'9 = Asin.= 9+(B-A)sin.'f;
.■.by equatiott 81,
U(«,0 = W(H.-H^Ters.a + |f.{Asin,'9 + (B-A)sin.'?}, .... (88),

5y which formula (he S/yna/mAcal stabiliti/ of the ship U represented, iotlt
as it regards a pitehing and a rolling motion.

If in ec^nation 31. i; = ", the line in which the plane PQ (parallel to the

deck of ■the ship) intersects its plane of flotation is at right angles to the
length of the ship, and we have, since in this case 9 = J (see equation 83,),

which expression represents the dynamical stability, in regard to a pitch-
ing motion alone, as the equation

U((l) = W(H, — HO-^-i

represents it in regard to a rolling motion alone.

10. If a gioen quantity of work represented by 0(9) be supposed to
be done upon the vessel, the angle e through which it is thus made to

roll may be determined by solving equation 8S. ■with respect to md-j-
Te thus obtain
■ .» W(H,— H,)+^A— -v/{W(H,— H,)+ftA}'— 2ftA.ir(9)_^^(8li-k

./Google

ON THE ROLLING OF SHIPS. 6i7

]7. If PE ana Q8 be conceived to be sti'aight lines, so that FOR and
QOS are triangles, then w. a, taken in respect to an element included
between the section OAD, and another parallel to it and distant by the
email sptice dx, is represented by

-fiy,y^sia.9d3^m^ + iiK);

4

by Jiiisin.''ay}ys3xe;

13

.■ . ws= —fism.'S f y]y^x,
12 ■'

and, eqnation 39

U(«,0=W(H,-H.) v««..+i ;.,iii.-»/j!!,,i, . . . (8!),

which fonnnla may be considered an appiroximate measure of the stahility
of the Tosael under all circumstances.

If, as in the case of the esperiments of Messrs. TVjniftw ajid Eawbosi,
the vessel be prismatic and the direction of the disturbance perpeudicular
to itsasis,

^1 = constant = ((, and 3= iasin.e;

tJ(fl)=W(H,— H^Tera.e+iaw sin.9.

A rigid surface on vMih the vesael mmj ie mppoied to rest wMlst in ihe
act of oiling

If we imagine the position ot the centre of gravity of a vessel ^oat
to be oontrnmllv changed b\ altenng tte positioaa ot some of ita con
tained weights without alteiu g the weight of the whole so la to ca ise
flie vessel to inchne mtn an it finite numl er <if diffeient positiona dis
pladng m each the same volume of water then will the different plan i
of flotation, corresponding to these difterent positions, envelope a curve-!
surface, called the surface of the planes of fiotatiou (an/y/aee dea Jlotaisons\
whose properties have been discussed at length by K. Dttpis in Ms es-
cellent memoir, Snr la Stability des Corps Mottants, which forms p«.t of
his Applications de G6ometrie.* So for as the properties of this smface
oonoern the conditions of the vessel's egvAUMmn., they have been ex-
hausted in that memoir, but the following property, which has reference

* BACHitLina, Paris, 1822.

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6-i8 APPENDIX.

ratlier to the conilltions of its dynamifial atabilitj than its equilibrium, ia
not stated by M. Dhpin : —

^ we eonaeke the surface of the plmies i>f flotation to lecame a rigid
swrfaoe, and oho the sur/ace of the fluid to lecame a rigid, plane without
frietion^ so that the former surface may rest upon, the latter and roll and
»Ude upon it, the other parts qf the «e»ael heing imagined to he so far im-
material as not to interfere with this inoUon, hit not so as to take away
their weight or to interfere uiith the application of the upward pressure of
the fluid to them, then will the motion qf the vessel, wJien resting iy this
earned surface upon tkie rigid but perfectly smooth horizontal plane, ie
the same as it was when, acted vpon hythesamefojve,it rolled and pitched
in the fluid.

In this genera] case of the motioa of a body resting by a curved sur-
feoe upon a horizontal plane, that motion may be, and generally will be,
of a complicated character, inolnding a sliding motion upon the plane,
and simultaneous motions i^ound two asea passing throngh the point of
contact of the surface with the planes and corresponding with the rolling
and pitching motion of a ship. It being however possible to determine
these motions by the known laws of dynamics, when the form of the
surface of the planes of flotation is known, tie complete solution of the
qnestion is involved in the determination of the latter mirfacB.

The following property*, proved by M. Dupia in the memoir before
referred to (p. 33), effects this determination ; —

"The intersection of any two planes of flotation, influitely near to each
ether, passes throngh the centre of gravity of the area intercepted upon
either of these planes by the esternal snrfiioe of tlie vessel."

If, therefore, any plane of flotation be taken, and the centre of gi'avity
of the area here spoken of be determined with reference to that plane of
flotation, then that point will be one in the curved aurface in question,
called the snrfaoe of the planes of flotation, and by this means any numher
of such pointa may be found and the surface deteiinined.

The amis about lehioh a vessel rolls may ie determined, the direction in
which it is rolling "being gveen.

If, after the vessel has been inclined throngh any angle, it be left to
itself, the only forces acting upon it (the inertia of the fluid being neglected)
will be its weight and the upward pressure of the fluid it displaces ; tie
motion of its wntre of gravity will thei'efore, by a well-known principle
of mechanic, be whollj in the same vertical line.

Let HE repieient this vertical line, PQ the surface of the fluid, and

nM3 the suitace of the pknea of flotation. As the centre of gravity G-

"traverses the vertical HK, this surface will partly roll and partly slide

hy its point of contact M on the plane PQ.

If we suppose, therefore, PEQ to he a section of the vcsaol througli

* Tliis property appeal's to hfive beeu first givea by Enij;K.

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OS THE EOLLIHG OF SniTS. 641)

llie point IT, and perpendicular to the axis about which, it is rolling, and

if we draw a vertical line MO through the point M, and tbroagli Q a

horizontal line GO parallel to the plane PRQ, tlian

the position of the asis will be determined by a line ■'''*''■ *

perpendicular to these, whose projection on the plane

PEQ is O.

!For since Ihe motion of tie point G is in the verti-
cal line HK, the asis nbont which the body is revoly-
ing pa^es through GO, which is perpendicular to
HK ; (aid 9inc« the point M. of the vessel ti'averses
the hue PQ, the axis passes also through MO, which
is perpendicular to PQ ; and GO k drawn parallel to, and MO in the
plane PEQ, which, by supposition, is perpendioulai' to the asis, therefore
the axis is perpendicular to GO and MO.

If HK be in the plane PEQ, whioli is the case whenever the motion »
exclusively one of rolling or one of pitching, the point 0 is determined by
the intersection of GO and MO.

The time of the rolUng through a, small angle of a veiael itihose athieart
seationa are (in respeet to the parts luJyeet to immersion and emersion)
dreular^ am,d hmie their centres in the same longit'odinal arxii.

Let EDF {flg. 1. or fig. 2„1 represent the midship section of such a
Fig.i.

vessel, in which section let the centre of gravity G, be supposed to be sitn-
otetl, and let HK he the vertical line traversed by G, as the vessel rolls.
Ima^ne it to hiive been inclined ftrom its vertical position through a ^ven
angle 9| and the foreea which so inclined it then to have ceased to act
upon it, so as to have allowed it to roll freely back again towai'ds its posi-
tion of equilibrium nntil it had attiuned the inclination OCD to the verti-
cal, which suppose to be represented by e.

Referring to equation I, page 132. let it be observed that in this ca-so
Stij=0, so that the motion is deteimined by the condition

SM,= ~-Ew'....(38).

But the forces which have displaced it ft'om the position in which it
was, for an instant, at rest are its weight and the upward pi'essure of the

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650 APPESDIX.

■water; and the work of ttese, U(ei) — tJ(8), iloiie between tie inclHiationa
e and fl, ivhea the vessel was in tte act of receding from tke vertical, was
ahown. to be represented by CW,fe,TTiAi) (vera, e — vers. 9,); therefoi'Q
the work, between the same inohnations, when the motion is in the
opposite direotioii, is represented by the same expression with tbe sign
changed ;

.-. rw,=(WA=F"W"A)(vers. fli — vers, e),

a,nd since the asis abont which the vessel is revolving is perpendicular to
the plane EDF, and passes through the point O, if Wik' i-epreaents its
moment of inertia, about an axis perpendicular to the plane EDF, and
passing through its centre of gravity G„

Substituting in. equation 38. and writing for OGi its value fti sin. S, we
(WAtWA) («r,. .,_«». .)= ^ (i-+J!.ii.'.) (I?) ;

\/»S.(l=FW5;) /

assuming s to be so small that the fourth and all higher powers ol
1. -^ Q may he neglected, and observing that, this being the case,

%/'■ "■' I •+*«""-i • = \/*'('+"'' 5 • )+* "•' 5 °

, Google

ON THE EOLLIXG OF ;

V'

s/'^^i^mJ sj^

a"' '^m-'a"

The sign + being taken according as the centra of gravity of tie displaoed
fluid aawnds or descends.

The time of a veseeVs rolling or pitching through a small angle, its form
mid Mmmsiom being any vihatev^.

Let EDF (figs. 1. or 3.) represent the midship section of audi a vessel,
supposed to bo rolling about an axis whoae projection is O; and let 0
represent the centre of the circle of curratnre of the sur&ce of its planes
of flotation at the point M where that surface ia touched by the plane PQ,
being above the load water-line AB in fig. 1, and beneath it in fig. 2. Let
the radius of ourvoture OM be represented by p ; then adopting the same
notation aa in the laat arlaole, and observing that the asis O about which
the vessel is turning is perpendicular to EDF, we aball find its momoat of
inertia to be represented by

w,i<.H(H.^,)..ta...!(|):

■where Hi repreaenta the depth of the centre of gravity in the vertical por-
tion of the vesaol.

, Google

652 APPENDTS.

Also, by equation 35.
Si[,=U(9,)--i;(»)="W",(H,— II,)(c<i3.S— cos.9,)+i-ftA(cos.'a— cos-'fl,),
.'.by equation 38.
ir,(H,-H,)(=....-...,.,)+|,A(„...-oo..-..)=| j F+(H,-,)..in... J (|)'

•w=^.

¥

Assuming 9 and «, to be so small that cos. e + oos. S, = 3, and observing

' /" '/y-^(H.-,)-iCT

~9l

moreover, p to remain constant ijetwcen tlie limits—^, ari'
ting aa in equation 89.

'TV — .'^ » 2 ' I ■ ■ '

Since tbe value of sin.'nS, is exceedingiy araall, tJie oscillations aiB

nearly tautocLronous, and the period of eacli is nearly represented by the
formula

y,(H.-H..^^)-

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KQUILIBRIUM OF PKESSCllES. OOd

Tie foUowing raetiod ia giyen by M. Dupis for determining tlie value
ofp*:—

" If tlie periphei'j of the plane of flotation be imagined to be loaded at
every point with a weight represented by the tangent of the inclination of
the sides of the vessel at that point to the vertical, then will the moments
of inertia Of that curve, BO loaded, about its two prindpal axes, when
divided by the area of the plane of flotation, represent the radii of greatest
and least oorvature of the envelope of the planes of flotation."

If p be taken to represent the radius of greatest onrvature, the formnla
41. wiU represent the time of the vessel's rolling; if the radius of least
curvature (B being also sabstitnted for A), it will represent the time of
pitching.

On tJie conditiom of tTie equilibrium, of (vn/y mmber 0/ presswes in the
»wme plam, applied to a iody moveabh ahont a eylindrical axis in the state
hordervag upon motion. (From a memoir on the Theoi-y of Mechanics,
printed in tiie second part of tlie Transactions of the Eoyal Society for 1841.)

Lot Pi, P„ Pj, &o. represent these pressures, and E their resnitant. Also
let (i|, Oj, flSj, repi-esent the perpendioulai-s let ftdl upon them severally from
the centre of the axis, those perpendiculars being taken with the positive
dgns whose corresponding pressures tend to turn the system in the same
direction as the pressure Pi, and those negatively which tend to turn it ia
the opposite direction. Also let % represent the perpendicular distance of
the direction of the resultant E fi-om the centre of the asis, then, since E
is equal and opposite to the resistance of fJie asis, and that this resistance
and the pressures P,, P,, Pj, &c. are pressures in equilibiiuni, we luLve by
the principle of the equality of moments,

P,((, + P,!^ + P,a, + &«. = JlE.
Eepreaenting, therefore, the inclinations of the directions of the pressures
P,, Pi, Pj, &c. tD one another by ■,„ 'u, i,,, +, &c,, &o., and substituting
for the value of E-t

* ApplicBtions de GdomStrie, p. 47.

\ The inttlinalaon 1,^ of the diraotions of any two pressures in the above ex-
Dression is taken on the anppoaition that both the pressures att from, or both
iomardt the point in whieh they interaeot, and not one toaards. and the othe»
from, that point; so that in the case represented in the figure in the note at p.
175,, the inclination i, , of the preeauraa P, and P„ represented by the arrows,
is not the angle P, IPj, but the'sngle P.IQ, sinoe IQ and IP, are directions of
these presBurea, both tending/rom this point of intersection, whilst the direo-
tions of P,t and IP, are one of them toviarde that point, and the othar from it

% FoissoH, MScsuiqne, Art. 33.

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+2P,P,coE, 'i^+SP.r^c
+ 3P,P,co*. .5^ + 3P.P,c

+&0. &a

Pi' + 2Pi(P,cos. (,., + PjCOs..„ +
+ P,' + P,' + P,'+ . . -

+ aP,P,003 + 2PsP.TOa. <.

+&e. &c.

If tie value of P, invol^^ed in this equation be expanded by I
theorem *, in a aeries ascending by powers of i^, and terms involving powers
above the first be omitted, we shall obtain the following value of that
quantity : —

_PA+P.a3+..

a.

^(Pa + Pa + P,^, + ....)'

Q

-i(P^ + P,a, + PA + . ...)

(PiCoa.i^+P^cos. <,.,+P.cn
+P,'+P^+P,'+ ....
+ aP,P,co.s..,,+2P,P,cos..,,
+ 2P,P,cos.i,..+....

s. „., + ....)

reducing,

P^+P,a

i±^ +

Pj'((ti' — 2a,aa 009. (,,1 + 1(5^
, +P.Xff.'— SaAoos-'ii+ais')

+ 2 P, Pj-fd/i^, — ff,(a, C0S.1I.S+ 0

ITow a? — 2aiajOoa, ",.,+05 represents the square of the line Joining tb&
feet of the perpendiculars a, and a, let fall from the centre of the asia
npon P, and P,; similarly oj — 2ii|«j cos. iij+olrepresenta the square of
the line joining the feet of the perpendiculars let fall npon Pi and V„ and

* This expansion may be effected by squaring Ijoth aides of tho equation,
solving the quadratit; in respect to P„ neglecting powers of > above the first
and reducing ; tiiis mctliod, however, is eseeedingly laborioaa.

, Google

BOLLIKG MOTION OF A CYLISDEE. Ooa

HO of the rest. J^t these lines be represented by L,.s, Li.„ L, 4, &o., and let
the different values of the function

{OjOj — &, {a, COS. tj^+(^co8. 1,.3+tSj Gog. t,,)}
be reproaented hy M,,, H,a, ^,-4, &o.

..p,= +1 ^ 3P,P,M,,+3P,P.M,.,-i-., . J ■•■■<*■'>•

B"OTE E.
E Eor.LrHG Motion ob

{iVom a memoir printed in the Transactions of the Eoyal Sodsty for
1851, part II.)

The oscillatory motion of a heterogeneoas ojlinJer rolling on a horizontal
plane has been investigated hy Eoibe.* He has determitted the pressure
of the cylinder on the plane at any period of tlie osoillatiou, and the time
of completing an oscillation when the area of oscillation are ^mall.

The forms under which the cylinder enters into tiie compod-tion of
machinery are so varions, and its uses so important, that I Lave thought it
desirable to estend this inquiry, and in the following paper I have songht
to include in the disouswon the case of the eontiuuoua i-olling of the cylin-
der, and to determine —

1st. The time occupied hy a heterogeneous cylinder in roUing continu-
ously through any given space.

2ndly. The time occupied in its oscillation through any ^ven arc.

8rdly. Its pressure, when thus rolling continuously, on the horizont^
plane on which it rolls.

tinder the second and third heads this discussion has a prftotical appli-
cation to the theory of the pendulum ; determining the time occupied in
the osoillatJona of a pendidum through any given arc, whether it rests
on a oyhndrical axis or on knife-edges, and the ciroumstances under
■which it will jump or slip on its healings ; and undet tlie first and third,
to the stability and the lateral oscaUaliona of locomotive en^nes in rapid
motion, whose driving-wheels are, by reason of their cranked axles, untruly
balanced.

* Nova A<^ta AeaJ. Petropol. 1788. " De motu osoillatorio oirea nxcm cyllit
diieiiiB plniio horizoutali iiicumbontem."

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058 APPENDIX,

I,et AlIB represent the section of a heterogeneous cylindci' tJiroi;gh
its centre of gravity G and perpenclienlEr to its asis C ; and let M be its
point of contact, at any time, with the hori-
zontal plane BD on which it ia rolling.
Assume

a = AO,h = GQ,S = AGM.
■W = wdght of cylinder. WF= nioinea-
tura of inertia of the oylindei' about
an asis passing tiirough G and
parallel to the asis of the cylinder.

w = given value of tlie angniar velocity ( -tt ) when fl has the given

valne fl(.
0, = given value of 6 when the angniar velocity hue tie given value a.
I = ^ven valne of GM corresponding to the vahie 8, of B.
Then W (P + Gtf) = yfO? + a' + ft' - 2oA cos. 0) = moment of inertia
abont M. Since moreover the oyKnder may bo considered to be in the act
of revolving abont the point M by which it is in contact -with the plane,
one-half of its m« ciua is represented by t]\e fovmnia

l>^

..-2.S0,..

-)(iy.

and one-half of the uis iiwc
«,-fl,hy

( aoq^nired by

it'

in rolling thraugh

the

angk

But the vertical descent of the centre of gravity while the cylinder is
passing from the one position into the other, is represented by
Atcos.9 — oos.e,).
Therefore, by the pnnciple of sis «Mia,*

l~\(¥+a'~%ahoos.ei'}f)(^)—(h''+l')a'\='Wh(ooa.B-oos.e{),

whence we obtain

/^y ^ agA(fl03.s~oo3. 9,) + (h'+P)^'
\dty *'+«" — 2aA COS. fl-nft.'

■■«'"='(5+s+i)---«-

' PoB^BON, Jiyttamique. 2"° partis, 565, ; Pokceiet, Mhamqiie IndiislHeUe,
•t Art. (129.) of this Worit.

./Google

ROLLING MOTION OF A CYLliSDER. hilt

where ( represents the time of the body's passing from the incUnatJoa 9i to

Sow let it be observed that in this function a.>i} so long as a is less tbaa
y.fflnee

l!' + ?>-(i?^l^^\ or i' + a' — '2ahcos.9, + K'>~(7^ + ^^\
and .■,P + a' + A'>2«Aooa.e,— (i' + !')"'')

•^^ st + A + aJ^"^^-"'-^!--"-

Then wlicn 9 = 0, q' sec' * = ^- =q\ ■'■ see. (S = 1 and ^ = 0.
When « = fl, lot # = ^„

sf— i+ - + iWcos.n,

a— GOS.fl. V'A « a;

also

_/?\ F+(«— A)'

J l,c,»-»-sJ J l,oo...-5J *^

Anil sine. -— F^ =

— (ci+B) _ eoa.' ^ — g'

(a— 6) ~co?.= * + 5"

(.+s;»..'f^rt+(.-,s)(co..->-rt

»■•'= (SiiTT^-r^

42

, Google

8111.' B =

(coa.'^ + g- — aco9.V — 3?')(co

■,j)g'+(l— »)cos.'>>}{(l+.B)^+(l+c.)e(

os.= rt'

?'+y'eo«.'tfi _ _ ^ ^ ^^^_

^_ (J5 ^cos. 9 (Jcos. li_Bin. -J daos.0
df d COS. e ■ d COS. 9 ' (i^ sill. 0 ' d oon. #■'"'■ ''
Also by equation (0.),

■ . by equations (T.) and (8.),

ds _2(a — .ay g' + co5.'ji COB. »

^#-^(1 — 13')*j ■(?'+ycos.^^Ji ■ (2'+^>j.-,\)'

■ (l-A (iZ'+eos.V)(2'+ii'cos.'^)i

/g — wa.'Yi'i^gfa. — g).;' 1 1

■ \M%.s—^)dp (1— (S-)'"' Uif+cos.V)fe"+ii'c
_ afg — 3>g' j 1 )

_3(g — J5V?^

-(l_^'ji(p" + 5"_li(l+^')

+ !W +<?')) ^ ,_J_ ., A/ ^ _ _J':^j,;„ v^s'

1 1 1—5 ,

l+,3 _ (l+a)a-/?)

./Google

KOLT.ING MOTION <

2'a — %' f d'P

"(1 — l!")'(j>'+}-)'(l+l'

K"->^

■ (11)-

■where Il( — ncji,) is that elliptic function of tiic third order wliose pur*
metfii' is — n nnd modulus o.

-'-= yi-j; ,._-!-

\/(a+I+i)-«"-''-

fe' + fa — ai'

.•.by eqniitlotis 11. and i.

p+[a— ay

where (9.) (3.) (3.)

0, + "S-s" 2ATC1-S. Oi + u"

•I").

• I catinni find that this funotion hns befora been intognited; esc^t ia the

e in which ;) is exeec lingly Bm;ill,

, Google

600 AP^^:^"^IX.

tra (10.) (8.) (3.)

Tlie value of n( — ncp,) Tjeiiiji; determinable by known methods (Leobk-
DRB, Fonctiona EUipliqnes, vol. i. chap, xxxiii.), the time of rolling is given
by equation 13.

In the case in which the rolling motion is not continuous but oscillatory,

we have a = 0; and therefore (equation 5.) fi'^-; n{ — JWiS) beuomes

therefore in this case a complete function.

To expi'ess the value of this complete elliptic function of the third order
in terms of funotions of the flrst and second orders, let

Thon-^

Eopresenting therefore the time of a semi -oscillation by t„

^^^V)

. . (18).

Since the values of elliptic functions of the first and second oi'dera,
having j^ven amplitudes and moduli, are given by the tables of Legbndke,
it follows that the value of ( is given by this formula for all possible valnes
offl and^..

If the angle of oscillation 0, be very small e is very small, bo that its
square may he aeglected in comparison with unity. In this case

• Leqendp^, Calens 'les Fonotions Elliptiqnps, vol. i. chap, xxiii. Art. 116.

, Google

BOLLING MOTION OF A CYLINDER.

Fi^ = Eo^ = ■;. £Lud Fc - =: Eo - =

.-.Fc^-'Eci—'Ea'^Yc^^O.

For small oscillations therefore

■ 09).

If tlic penduiuiii oscillate on knife-edges a = 0, ; = /t^ and we oljtain tlie
well-kiioivn tlicorciin of Legendbe (Fonctiona Flliptiques, vol. i. chap.

rill.)

- (30)-

In the case of the small ciscilktions of a pendulum resting on knife-edge,
equation 30. becomes

t= /*!4l.«....(23),

which is the ivell-lmown formula applicable to that case.
If the pendulum be one which for small arcs beats secoada (21.),

by which equation the time of the oscillation through am/ are, of a pen-
dulum which oscillates tlirongh a small arc in one second, may be deter-
mined. I hare caused the following table to be calculated from iL

, Google

002

Tu,li!e of tlie tj

AVPKNDIX.

CQUpied in oscillating throngii every two tlegi'eea
ly a pendulum wMch. Oicillates tbrougli a small a:

A>^rf™illnlBll

Ti™or™B

A., tf^icilkllon

n„c«r™»

Aro otMrlJkiiijn

Ti,n! of oiiB

'"■"'••"■

"°°""-

'"*'""■

■™"""-

'"'^^'

•"""""■

93

1'1899

122

1-3905

162

1-8033

94

1-2U0I)

124

1-4U39

164

1-847B

96

1-21-2S

156

i-aari3

I'^ilO

144811

153

100

i-is-i±

1 -ilillS

T6U

102

1 2439

132

l-49i-i

1112

104

IBt

1-5157

164

2-1 453

]{I6

1'2U3G

136

1-S4U0

2-2283

108

1-2817

138

1-6067

168

2-3248

no

1-2HE3

UO

1-0944

170

2-43B3

112

1-3099

U2

m

2-58111

r3'>4-9

144

1 -uasi

174

1-3-iOO

146

1-H8S1

176

fi-(il9S

1-3360

148

l-IUO

173

3-4600

lio

13129

ISO

1-7622

5 80

Infinite.

The pressure of the eylindm' on its point of contact with the plane on lihich

Let A' be the point where the point A of the cyllndoi- was in contact
with till,' plane.

Let A'N = «, Fa = y.
f ^ ^\ — X = horizontal pvessnres on It in tlireotion

f ^ o I A'M.

= vertical pressure on M in tlireotion MO.

the centre of gravity G moves as it would

being collected -tliere, all the impressed forces -vvei-H

do if, the -^vliole
applied

t, we have, by tJie pi-inciple of D'Alembeht,

9 dff-

Y-

-wi

«,0G

= %

MOA

a = a9-

-ft

in.e,

y = a-

-ftc

03.8;

^^.(.

-lie

, Google

MOITON OF A CTIJHDEK.

'$)*"-

Assume ( — J =11,

W

. ■ . by equation (39.),

N(<t— Acos.e)
mcos.9— NAsin.u;

?j-l[.ta^.+N(»-„,..)|j

But by equation (I,), anbstitnting — 8 and — 0, for S atid D„

,^, 2aft(003. e — COS. fl,) + (F + JO -

p+„'+7i'-2uAoo9.ff, + (S'+r)— — (i'+(t'+A-— 2uSec

=©■

■ Vay ( i?+tti+s,'_2ttftcos.e ) " "
Observing that o'+A'— SoA cos. e, = P.

PLffarentiating this equation and dividing byf -=- Jj

./Google

664 APPENDIX.

Substituting these values of M and N in equation (80.), aiwl roilnclng,
__ WhAn.S I (F + P)(i,- + A'_gAros.flXg4-^>o') > ,„ .,

The rotation of a body about a eyliiidrical am« of small diameter.
AsaumiDgo = 0 in equations- (81.), (33.), and ej=0, we have
2g^0O3.fl— 1) , -^ g^ sin, n

Thei-eforo, l)y eqnsition (30.),

^WA(tfA(3-

-^—«" kin. 9.... (40).

g j F + S^

^ "^ ^ Ti ^^^"5- -^" '=''^' "[■■■■ t-*^'-

The lost equation may be placed unJer tlie forra

''=^+TTF|l''-+i(w"-')t-s(w'->)-ii-

If--( -^v- u' — 1 I be mnncrioally less than tmitv, ivhetlier it be positive

or negative, there will be some valne of e between 0 and it for which this
expression ■will be equalled, with an opposite sign, by cok. b, and for which
the first tenn under the bracltet in the value of Y will vanish. This cor-
responds tu a minimum valne of Y reijresented by the fominln

if y^!±^V— 1\ be numerically gi'eater than nnity, then the
3\ ^?^ /

of Y will be attained when e = st, and when
(48).

it'' I

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ROLLING Kl-iTON OF A CYLIKDEE.

Tin Jump ijf an Axis.

If Y lie negntive ia any position of tte body, the asis ivill obviously
jump from its benrings, unless it be retwued by some mechanical espe-
dient not taken account of in this calculation. But if Y be negative in
any position, it mast be negatiTe in that in which its \ alue is a minimum
If a jump take place at all, therefore, it will take place when Y la a mim-
murn; and whetlier it will take place or not, la deteimined by finding
■whether tlie niinimum value of Y 13 negative If theieloie tlie esprebsion
(43 ) 01 (48 ) be r^ative, the axia will jump in the corresponding oa^
An axia ut infinitely aivmll diameter such as we have here supposed
becomes a fixed axis , and the pleasure upon a fixed axis, sapposed to
turn in ojhndricoi bearings without /ncUon, la the same, whatevei mav
be lis diametei , equations (40.) and (41 ) detenmne therefore that pies-
Buie and equation (42 ) or (48,) detenmnea the vtiboal stiain upon the
eolhr when tht tcndtnoy of the axi'' to ]i up fiom ica leai ngs is thi,
feKateat

The Jump of a Boiling Cylinder.

Whedier a jump will or will not take place, has heen shown to bo dotei--
mined by finding wiiotlier the niinimmu value of Y be negative or not.

Subatituting a for-/— ,4--+|| and reducing, equation (86.) becomes

ff/l-

_W(7.' + ;')(ff + au')S^_

S=w)^J-^g^3^^J}coa..^gJ^^2^W..

.■.5=.0, l«t,whcn^-'i:iy+tl'^1^^^=0' 2udly,when.=«,

at « "ly"' l" — coo. 1)

8rdly, irhen s=0.
The first condition evidently yields 3 positive value of -3^, since il

, Google

eCfl Ari'ENDIX.

oaiises tlie first term of tte preceding equation to vnnish ; and the second
teiia is eBaentiallj positive, c being nhvays gi'eater tlian unity.

If, therefore, tlie first condition be possible, or if tliere be any value of
fl whiiih satisfies it, tliat value oojTeaponds to a position of n
sure. Solving, in respect to cos. e, we obtain

-</'-

{!r+l'%g+a^-Ha.'—l)_

■ (")■

The first condition ivill tlmtefore yield a positi
Bure, 'if

'/(i'+fXln-i.'XJ=l) >-l „„'/W±l

-V ii-i < + 1, "" V

(f + PXgt»')(»-l)

(»+!■)»+»

^•X.+I)

sgm-

-IF

, ^ S|7fc

+ 1)'

SgM,-

ilKa-ir

»^ "' (if+rX. + l)'"'" '(f+P)(.+l) i'
wlience, snbstitnting for a nnd i!p(lucing, we obtain finally, tbe conditions

Of tbeae ineqnnlitics the second always obtains, becanso
whatever be the values of h, a and h, And the first ia always possible,

If the^™( obtain, there are two correspoiiding positions of OA on eitlier
ride of the vertical, determinftd by equation <46.), in whicli the pressure Y
of the cylinder upon the plane is a minimum.

, Google

KOr.LING MOTION OT

', the other two Tiilues (:( iind 0) of 6 whioli tiiuse — U

vanish in tha Tulue of -j-j we ohtain tie Yaluca

~U — ^gJii:+ir^\'"'^U mi^if — i'

or

■which eipreasioii3 are both negntive if the ineq^nalities (47.) obtiun. Tie
same conditioua which jield minimum values of Y in two corresponding
oblique positions of OA, yieM, therefoi'e, nwsimum values in tlie two vei'-
tical positions ; so tiat if the inequalities (48.) obtain, there are two posi-
tions of maxnnma and two of aiiiiimnin pi-essure.

Substituting the values of eos. s (equation 46.) in equation (44.), and
reducing, we obtain for the mirnmum value of Y in the case in whioh the
inequalities (4ft ) obtiun,

+ » /(i'+l)if+(.+;.) i ji' + (a-i)'l h+j)\-

If this expression be negative the cylinder will jump.
In the caaein which a=:Q, which is that of apeiiduhuiihav'mga cylin-
drical axis of finite diameter, it becomes

J^L\ Sa'— 2A'— 3F— Z'+3 °V(h'+P){ii-+(fl+hy\{¥+(a-ky}l''....l,
id't '

If the flrat of the inequalities (43.) do not obtain, no position of mini-
mum pressm-e oori'esponds to equation (46.) ; and the inequalities (47.) do

not obtain, so that the values (49.) of -tt, ^ven respeotiTely by the sub-
stitution of rt and 0 for 9, ai'e no longer both n^ative, but the second only.
In this case the value n of « is that, therefore, whioh cori'espoiids to a posi-
tion of minimum pressure, which minimnm pressure is determined by
substituting n for 9 in equation (36.), and ii

• Wlien the pendulum oatillat^s on knife-eilgas a=0, and this expi-esaion
assumes tlie form of ft Yaiii«hing fnction, whose value may be determiued by
the known rules. See tlie ne\t urtula

./Google

= -- Ja + A--?

to+<>«-) J

.■.Y=W<l-'^ + ^ LI >..,,(M).

The cylinder will jump if this espn?3sion he negative, that is, if

t, substituting and redneing, if

4A(»-l-A)cc

h}l-i

¥+1' " )

If the ftngular Telocity u be assumed to be that acquired in the liig
position of the centre of gravity, 01=71, and COS. rfl, = 0. In this i
therefore (equation 51.)

T=w(l^^");...(5a,.

and there will be a jump if u'> ? . . . (58}.

Tlte Pendulum oscillating on Knife-edffes.

In this case a is eyanesceiit., aud u=0. Equations (31.) and
become, therefore,

t + A' i'+A'

Substituting these valnes of M and H" in equation (80.),

wy I _ _ J

.■.X = -J^i,(.co.^-3eo...).in.....(5.).

./Google

ROLLING JIOTIOX OF A CVLiSnEK. I}(j9

k' + hXh' 3 J

There will therefore be. a junip of the pendnlnin upon its bearings at
eadi osdlLition if the amplitude e, uf tiie oscillation be sudi, that

27ie Jiiinp of tJie fahely-halaneedi Garriage-wheel.

The theory of the falaely-bnlsuiced carriage-wheel di&ors from that of
the roiiing oyliiider, — 1st, in that the inertia of the carriage applied at its
asle inflaence3 the accelemtioa prodnoed by the weight of the wheel, as
ita centre of gravity deacenda or ascends in rolling; and, 2ndly, in. that
the wheel is retained in contact with the plane by the weight of the car-
riage. The first cause may be neglected, because the displacement of the
centre of gravity is always in. the carriage-wheel very small, and because
the angular velocity is, compared with it, very great.

K W| represent that portion of the weight of the carriage which must
be oveiTjome in order that the whee! may jump (which weight is supposed
to be borne by the plane), and if Y, be taken to represent the pressura
upon the plane, then (equation 53.)

r, = W,-)- Y= W, -f- W (l —'''"') (57).

In order tlitit there may be n. jnnip. this expression ranst be negative.

*>5(>40..

, Google

The Brking- Wheel of a Locomotive Engine.

Tlie atfention of en^neera was some yenra sinus tiireoted to tl:e effeots
ivliioli might result from, the false balancing of a wheel by accidents on
railwayi*, which appeared to he occasioned hy a tendency to jump in the
driving-wheels of the engines. The craiilced axle in all cases desti-ojs the
balance of the dnping-wheel unk^^ a counterpoise be applied ; at that time
tliere was no counterpoise, and tlie axle waa so cranked as to displace the
centre of gravity move than it does now. Mr. Gbobsb Hbato:s, of Bir-
mingham, appears to have been principally instrnmental in causing the
danger of this fal.''fr-balaiicing of the <lriving-wheela to ho understood. By
means of an ingenious apparatus*, which enabled him tn roll a falsely-
balanced wheel round the circumference of a table with any given Telocity,
and to mate any required displacement of the centre of gravitj, he showed
tlie tendency to jump, produced even, by a very small displacement, to be
so gi'eat, aa to leave no donht on the minds of practical men as to the
danger of such displacement in tlie case of locomotive engines, and a coun-
terpoise is now, I believe, always applied. To determine what is Uie
degree of accuracy required in snob a oonnterpoise, I have cnlculated from
the preceding formula that dii-plncement of the centre of gi-avity of a
driving-wheel of a locomotive-engine, which is necessary to cause it to
jump at the high velocities not unfreqnently attained at some paila of tlie
journey of an express ti'uin ; from sueli information as I have been able to
obtain as to the dimensions of sucli wheels, and their weights, and those
of the enginesf. The weight of a pair of diiving-wheels, si-i feet in
diameter, with a cranked axle, vai'ies, I am told, from H to 3 tons; and
thatof an engine on the London and Birmingham Eailway, when filled with
water, from 30 to 25 tuns. If n represent tlie number of miles per hour
at which the engine is ti'ave)Mng, it may be shown by a simple calculation,

that the angular velocity, in feet, of a six-fest wheel is represented by -- '

or by in very nearly. In this case we have, therefore, — since "Wrepivssents

the weight of a single wheel and its portion of the asle, and W, represents
the weight, exclusive of the driving-wheels, which must be raised tliat

" This appnratus was exhibited by the late Professor Cowpee io illustrate hia
Leoturea ou Maohinaiy at King's College. It taa also been placed by Geiier;d
MoHDj among the apparatus of the Consevvntoire das Aits et Mi5tiei'3 at I'livis.

t I have not included in this eoioolntion tbeiuertin of the craak rods, of the
glide geaiing, or of the piston and piston rods. Tiia a£Eeot of tiiese is to ineresM
the tendency to jump produced by the diaplaeement of the centre of gravity
of ths wbeei; and tlie like elfeet is dne to the thrvist upon the piston rod
The discussion of tbcsB subjeeta docs not belong to my present poper.

, Google

MOTION OF A CYI.IKDEK. fj i 1 .

either sida of tbe engine may jnmp*, that ia. half Ihe weight of the engine
esolusiveof the driving --wheels, — ^=1^ to li tons, W, = 8tto Hi tons,

o =in, 3=82-!l0O84 whence I have made tiie following calculations f ram

foMimla (59.).

t!!B eiielije In

Sngtba'driT-
i„gwheeii.

^ >»ii(i.J)

or s di-feet rti'iriiiir--whef>l wliloh

—-'=«."— ■'"

.....,„.,„.„. ™..,.,h..,.

'

», j ...

TO,

»

,

41 2S 1 'asuT
■S43t -a^jsi

■iTgi

.a

3-5
3

ll5i -5150 ;«,

^~ -iaBa ■2908
1 1

■21B0

It appears, hj formula (59.), liftt the displaeement of the cent™ of
gravity neceaaaiy to produce a jump at any given apeed, is not dependent
on the aetaal weight of the engine or the wheels, but on the ratio of their
weights ; and, from the above table, that when the weight of the en^ne
and wheels ia 6J times that of the driving-wheels, a displacement of 3i
indies in the centre of gi-avity ia enough to create a jump when the train
ia travelling at sisty miles an hour, or of two inches when it ia travelling
at aaventy miles; thia displacement vai'ying inversely as the squai-e of the
velocity is leas, other things being the same, as the square of the diameter
of the wheel ia less ; for the radius of the wheel being represented hy a,

tlie angular velocity is represented byw=:— -, and substituting tliis value,

formula (B3.) becomes

nj\ y+w)

, Google

Dl2 APrKNDIX,

If the weight W of the whee! be sappoaed to vaiy as the square of iB
diameter and be represented by /m', this formula will become

-©'^.

still showing the displacement of the centre of gravity neeessaiy to pii>"
duce a jump to cUminish with the diameter of the wheel. These concla-
Bionsave opposed to the use of light engines and sninli driving-wheels;
and they show the necesaity of a careful attention to the trno balancing of
the wheels of the carriages as well as the driving-wheels of the engine.
It does not follow that every jump of the wheel would be high enough to
lift the edge of the flange off the rail ; the determination of the height of
Uie jump involves an independent mvestigation. Every jump neveithelesa
creates an oscillation of the spnngs, which oscillation will not of necessity
be completed when the jump returns bnt as the jumps are mode alter-
nately on opposite sides of the engine, it is probable that they may, and
that after a time they will, so sjnohioni^e with the limes of theoscillations,
as that the amplitude of eaoh oaeiUatiou shall be increased by every jnmp,
and a rocking motion be ccmmunicated to tlie engine attended ■with

Whilst every jump does not necessarily cause the wheel to mn off the
rail, it nevertheless causes it to dip upon it, for befoi'e the wheel jumps
it is clear that it must have ceased to have any hold upon the rail or any
friction.

The S%> of the Wheel.

I( f be taken to represent the coefficient of fllction between the snrfoce
of the wheel and that of tlie rdl, the actual frictioD in any position of tlie
wheel will he represented by T,/. But the friction which it is necessary
the ml should snpply, in order that the rolling of the whsel maybe main-
tained, is S. It is a condition therefore necessary to the wheel not alip-
jAng that

Y,/>S,or/>5....(eO).

If, therefore, taking the

irtain that the whee! cannot have slipped in that
1 the other hand,/ falls short of it, it must liave

aition for jumping, it is in an unfiivouriible position ot
It can only jump on one side nt once, and tlie efforts o

, Google

L. CYLTNDEK.

slipped.* The positions between which the slipping will talte place eoii-
tinnallj, ai'e detei-inined by solving, in respect to cos, fli, the equation

The application of these principles to the slip of the earrioge-'wheel ia
rendered less diffiottlt by the fact, that the value of h is always in that case

so small, as compared with the yaluos of k and a, ttat - may be neglected

in formulie (34.) aad (36.), as compared with unity. Those equations
then heuome

g (S' + a=) f

■ (63).

. ^M_l+gcoB.e a'M_{— )3(3+cos.fl)+a(14-3oos,e)}.Bi
" (SJ (B+COS.S)' di'' OJ+eoa.fl)=

Now if (3>1, there will be some value of fl for which g +

therefore 1+3 cos. S = 0; and since for this value of ^,-^ = 0,a]id

= 0, and

i(Pw

• Of couraa, the slipping, in the case ot ths driving-wlieela of a locomotive,
is diminialied by the fact;, that whilst one wheel is not biljng upon the rail
the other is.

43

, Google

= — -ei,_i-.iit follows that it corresponds i

therefore of^.
T,

But if |3<1, tten there is Bome value of cos. fl for which |J+co3.fl = 0,
and therefore for which w^iiiflnity, which value correaponds therefore in

this oaae to the

Thus then, it appenrs that aocording as

nvaliieofyia attained when cos. e=^3 or=: — ^; that

a value of y will he infinity, .... {&!).
mid in the other case it will be represented by the formula

X__1L_B(J!+£)1„. .i.(6S).

In the first case, i. e. when /3 < 1, the wheel will slip every time that it
revolves, whatever may he the value off. In the second cise, or when
|i > 1, it will Blip if /do uot esceed the number represented by formula
■(68.). The conditions (65.) are obviously the same with those (59.) which
determine whether there be a Jump or not, which agreei with an obsei"-
vnlion in the preceding article, to the effect, tliat as the wheel must cease
to bite upon the rail before it can jump, it must nlwaj's ahp before it
can jump. When the conditions of slipping obtain, one of the wheels
always biting when the other is slipping, and the shps of the two wheels
alternating, it is evident that the engine will be impelled forwards, at
certain periods of each revolution, by one wheel only, and at others, by
the other wheel only ; and that this is true iiTespective of the action of
the two pistons on the crank, and would be true if the steam were thrown
off. Such alternate propulsions on the two sito of the train cannot bnt

, Google

DE1^,^^^ [PON I"j linilD rn.'vr ' t

communicate ilterrate owillations to the buffer sinn,^ the irteivuls
between which wil! not bo the same as those between the pojiulsions
but the> laay so HyDchronise w th a serie of propnlaions as that tha
amjhtude of eich o'«iilktioB may be moreasel Ij them until the tri n
attamfl thit flsb tail motion with which railway tra^elkrs aie finnhai
Itia ibvious that the results lb ran here tflfoUo« ftom a disjlaceraent cf
the centi es ot gravity ot the driving wheels cannot tail also to be pro-
duced by the alternate action of the connaolang rods at the most favorable
diiving poinfB of the orank and at the dead points,* and that the operation
of these two causes may tend to neutralise or may esaggerate one another.
It is not the object of this paper to discuss the question, under this point

K DeSOEBT TtPOK AS IHOLISBD PlANB OF A BoDT SUBJBOT TO VaBIA-

Tiose OF Tempebatube,' and ow the Homos' os Glaciers.

conceive two bodies of the same foi-m and dimensions (ouhes, for
), and of the same material, to be placed upon a uniform horizon-
tal plane and connected by a substance which alternately extends and
contracts itself, as does a metallic rod when subjected to variations of
temperature, it is evident that by the extension of the intervening rod
each wil! be made to recede from the other by the tame distance, and,
by its oonti'action, to approach it by the same distance. But if they he
plaoed on. an inclined plane (one bang lower than the other) then when
by the increased temperature of the rod its tendency to extend becomes
sufBcient to push the lower of the two bodies downwards, it will not have
become sufficient to push the higher upwards. The effect of its exten-
sion will therefore be to cause the lower of the two bodies to descend
whilst the higher i-emains at rest. The converse of this will result from
contjaofion ; for when the contractile force becomes sufficient to pull the
upper body down the plane it will not have become sufficient to pull
the lower up it. Thus, in the contraction of the substance which inter-
venes between the two bodies, the lower will remain at rest whilst the
upper descends. As often, then, as the expansion and contraotion is
repeated the two bodies will descend the plane until, step by stop, they
reach the bottom.

A slip of the wheel may thus be, aud probably is, produced at eeoli ri

, Google

APPENDIX.

Suppose tlie uniform bar AB placed on
to extension from ii

nclined plane, and subjeist

330 of temperature, a por*

^^-^-^ tion 5B will descend, and the rest XA will ascend;

^ j;**^ the point X where they separate being determined

^ by the condition that the force requisite tfl push

XA wp the plane is eqnal to that required to push

XB down it.

Let AX = a, AB = L, weight of each linear tuut = ft, t := inclination
of plane, f = limiting angle of resistance.

.■,ftJ! = weightof AX.

^L— a!) = BX.

Now, the force acting parallel to an inclined plane which is necessary

to push a weight "W up it, is repre'iented by "W" — '-^—t ; and that ne-

oessary to push it down the plane by w _ ■ ■■ ■. {Art, 241.)

i.(^*+0

^ = ML-)'

a.{^-r)

fl!{flin.{i»+O+sin.(^—0]- = L sin. {}>—()
2»sin.^cos.i = Lsin.(^— 0

^^i,-L\\~

"When contraction takes place, the c
the above will be true. The separating point X
will be such, that the force requisite to pull SB v^
the plane is equal to that required to pull AX
down it. BX is obviously in this case equal to AX
in the other.
the elongation per Enear unit under any variation of tempera-
6 ; then the distance which the point B (fig. 1.) will be mode to descend
this elongation = ?,.BX

=^(L-:e)

Let

=»^(^^-*iS^)

, Google

: UPOX INCLINED I

If yre oonceive the bar now to return to its former temperature, con-
tracting by the same amount (j.) per linear unit ; then the point B
(fig. 2.) will by this contraction be made to ascend through the space

Totd dMoent Z of B by elongation and contraction is therefore determlued
by tie equation

To (Icteimme the pie^aure upon a mil driven
tliioUnh tht) rod at any point P fastening it to the
I lane

It is evident, that m the act ot extension the part BP of the rod -will
descend the plane and the i.ait AP astend, and cunvtrsoly in the act of
contraction ; and that in the former case the nail B will sustain a pressure
upwards equal to that nece^ary to cause BP to descend, and a pseasure
downwards equal to that necessary to cause PA to ascend ; so that, as-
Bunung the pressure to be downwards, and adopting the same notation as
before, escept that AP is represented by p, AB by a, and the pressure
upon the nail (assumed to be downwards) by P, we hare in the ease of
eatansion

in-(^0

■8 of contraction,

m.(^+0

P=^a— p)-
Kodudng, these forraulte become respectively,

My attention was first drstwn to the inflnenee of variations in tempera-
;ure to cause the descent of a lamina of metal resting on an inclined plane

, Google

678 APPENDIX,

by ub'ic.i'ving m the autumn of 1838, that a portion of the lead which
i-ovei6 the south side of tie choir of Bristol Cathedral, which had heen
renewed in the year 1861, bnt had not been properly fastened to the ridge
beam, hdd descended bodily eighteen inches into the gutter ; so that ii
plates of lead had not been inserted at the top, a strip of the roof of that
length would hn've been left exposed to the iveathet. The sheet of lead
■which had so descended measured, from, the ridge to the gutter, 19ft. iin.,
and along the iidge 60ft. The descent had been continually going on
fiom the time tho lead had been laid down. An attempt made to stop it
hydn^mg nails through it into the rafters had foiled. The force by
■nliich the lead bad been made to descend, whatever it was, had been
found sn&cient to draw the nails.* As the pitch of the roof was only
] G^' it was sufficiently evident that the weight of the lead alone conld not
ha¥e caused it to descend. Sheet lend, whose surface ia in the state of
that used in roofing, will stand firmly upon a surface of planed deal when
inclined at an angle of 30°+, if no other force than its weight tends t*
Ottiise it to descend. The considerations which I have stated in the pre-
ceding articles, led me to the conclusion that the daily variations in the
temperature of the lead, exposed as it was to the action of the sun by its
southern aspect, could not but cause it to descend considerably, and the
only question which remained on my mind was, whether this descent
could be so great as was observed. To determine this I took the follow-
ing data: —
Mean daily variation of temperature at Bristol in the

month of August ; assumed to be the same as at Leith

(Kcemtz Meteorology, by Talker, p. 16.) . - . 8° 31' Cent.
Linear expansion of lead through 100° Cent. - - - -0028436.
Length of sheets of lead forming the roof from the ridge

to the gutter ,...-.- 333 inohes.

Inclination of roof 16° 82'.

Limiting angle of i-esistance between sheet lead and deal - 30°

Whence the mean daily descent of the lead, in inohes, in the month ol
August, is determined by equation (2.) to be

* The evil was remedied by placing abeam aoross the rafters, near tharidge,
and doubling the Bheets round it, and fixing their ends witli spifee-naila.

■(■ This may easily be verified. I give it as tiie result of a rough esperiraent
of my own. I urn not aeqiwunted with any experiments oq the friotion of lend
made with airiSoient oai'e to be received as authority in this matter. Tlie
friction of copper on oat haa, however, bean determined by General Mobjn
(see a table in the preceding part oftbiswort} to be 0'63, and its limiting angle
of resistanee 81° 48'; so that if the roof of Bristol Cathedral bad been inclined
at 31° instead of 18°, and had been covered with sheets of copper resting on
oak boards, instead of sheets of lead resting on deal, the sheeting would not
have slipped by its vinyht only.

, Google

DESCliST rrON INCLINED PLANE.

=■027848 inches.

This average daily descent gives for the whole raoiitli of August a descent
of -868388. If the average dwly variafion. of temperature of tlis month
of August had continued throughout the year, the lead would have
descended 10'19148 inches every year. And in the two years from
1861 to 1853 it wonia have descended 20-38396 inches. But the daily
variations of atmospheric temperature are less in the other months of the
year than in the month of August. For this reason, therefore, the cal-
culation is in eseess. For the following reasons it is in defect: — 1st.,
The daily variations in the temperature of the lead cannot hut have been
greater than those of the surrounding atmosphere. It must have been
heated above the surrounding atmosphere by radiation from the sun in
the daj-time, or cooled below it by radiation into space at night. 2ndly.,
Om variation of temperature only baa been aiwmned to take place every
twenty-four hours, viz. that from the extreme heat of the day to the
extreme cold of the night ; whereas such variations are notoriously of
constant oocuri-ence during the twenty-four hours. Each cannot but have
caused a corresponding descent of the lead, and their aggregate result
cannot but have been greater than though the temperature had passed
uniformly (without oscillations backwards and forwards) from one extreme
to the other.

These considerations show, I think, that the causes I have assigned are
sufBcient to account for the fact observed. They suggest, moreover, the
possibility that reanlta of impoi-tance in meteorology may be obtained
from observing with accuracy the descent of a metallic rod thus placed
upon an inclined plane. That descent would be a measure of the a^re-
gate of the changes of temperature to which the metal was subjected
during the time of observatioa As every such change of temperature ia
associated with a corresponding development of mechanical action under
the form of work,* it would be a measure of the aggi-^ate of such changes
and of the work so developed during that period. And relations might be
found between measm-ements so taken in different equal periods of time
— successive years for instance — tending to the development of new
raeteorolo^cal laws.

* Mr. Joule has shown (Phil, Trans., ISEO, Part I.) that the quantity of heat
capable of rwaing a pound of water by 1° Fab. wquires for its evolution 173
units of work.

, Google

The Dmoent of GLiOiEiiS.

The following are tht results of recent experiments * on the expansion
ofi«e:—

Linear Expwnsion of Ice for an Interval of 100° of the Gentigrade

0-00534 Scliuraacher.

0-00613 Polirt.

000518 Moritz.
Ice, therefore, has nearly twice the expansibility of lead ; so that a
sheet of ice would, under similar circumstances, have descended a plane
rimilarlj inclined, twice the distance that the sheet of lead referred to in
the preceding article descended. Glaciers are, on an increased scale,
sheets of ice placed upon the elopes of mountains, and subjected to
atmospheric variations of temperature throughout their masses by varia-
tions in the quantity and the temperature of the water, which, flowing
from the surface, everywhere percolates them. That they must from this
cause descend into the vaWejs, is therefore certain. That portion of the
Mer de Glace of Charaouni wliich extends from Montanvert t« very near
the origin of the Glader de L^chaud has been accurately observed by
Professor James Forbes.t Its length is 22,900 feet, and its inclinAtion
vao-ies from 4° IS' 23" to 5° 5' 53". The Glader dn Geant, from the
Tacul to the Ool du Geant, Professor Forbes estimates (but not fram his
own observaUons, or with the same certainty) to he 34,700 feet in length,
and to have a mean inclination of S" 46' 40''.

According to the observations of De Saussure, the mean daily range
of Eeanmur's thei-mometer in the month of July, at the Ool du Geant, is
4°'257t, and at Ohamouni 10°'092. The resistance opposed by the
nigged chawiel of a glacier to its descent cannot but be different at dif-
ferent points, and in i-espect lo different gladers. The following passage
from ftofessor Forbes's work oontwns the most authentic information I
am able to find on this subject. Speaking of the Glacier of la Brenva
he says : — " The ice removed, a layer of fine mud covered the rook, not
composed, however, alone of the clayey limestone mud, but of sharp sand
derived from the granitic moraines of the glacier, and brought down with
it from the opposite side of the valley. Upon examining the face of the
ice i-emoved from contact with the rock, we found it set all over with
sharp angular fragments, from the size of grains of sand to tliat of a
cherry, or lai'ger, of the same species of rock, and which were so fivmlj

" Vide Arehir. f. WiaaenBoliaftl. Kuudo T, Eusslsnd, Bd. vi
f Travels through tlie Alpa of Sav"y. Edinbnrgli, 1853,
t Quoted by Piofesaor Fokees, p. 231.

, Google

DESCENT OF GLACIERS. 681

fised in the iee us to demoiiatrate &s impoasiWlity of suoh a siu'fece being
forcibly urgeil forwai'ds witliout sawing any compwolJvely aoft body
■whioli might be below it. Aoooi'dingly, it was not difficult to discover in
tlie Uinestone the Tery gi'oovea and Boi-at«bea wMcb were in tlie fiut of
being made at the time by the pressure of the ice and its contiuned frag-
ments of stone." (Alps of the Savoy, pp. 203 — 4.) It is not difficult
from this description to account for tlie fact that small glaciers are some-
times seen to lie on a slope of 80° {p. 85,). The most probable supposition
■would indeed fix the limiting angle of reristanc« between the rock and
the Tinder surface of the ice set all over, as it is described to be, with
particles of sand and small fragments of stone, at about 30°; that being
nearly the slope at which amootb, surfaces of oaloareons stone will rest on
one another. If we take then 80° to be the limiting angle of resistance
between the under sttriaoe of the Mer de Gla«e and the rock on which it
rests, and if we assume the same mean daily variation of tewiperature
(4-SS7 Eeaumnr, or C-321 Centigrade) to obtaia thi-oughout the length
of the Glacier du Geant, which De Sanssnre observed in July, at the
Ool du Geant ; if, ftirther, we take the linear expansion of ice at 100°
Oentigrade to be that (■00534) whioh was detennined by the experiments
of Schumacher, and, histly, if we assume the Glader de Geant to descend
as it would if its descent were unopposed by its oonfliienue with the
Glacier de Lecliant; we shall obtain, by substitution in equation (3.)
for the mean daily descent of the Glacier du Geant at the Tacul, the
formula

■00534 tan- 8° 46'

7= 1-8395 feet.

The aotnal descent of tlie glacier in the centi-e was 1'5 feet. If the
GUwier de LSchaut descended, at a mean slope of B°, singly in a sheet of
uniform bi-eadth to Montanvert without receiving the tributary glacier of
the TalSfre, oi- uniting with the Glacier du Geant, its diurnal descent would
be given by the same foi-mula, and would be found to be -05487 faet.
Reasoning similai'ly with reference to the Glacier du Geant ; supposing it
to have continued ih coai-se ^ngly from the Ool du Geant to Montanvert
without confluence with the Glacier de L^ehant, its length being 40,430
feet, flfid its mean inohnation 6° 53', its mean iTiuraal motion I at Montan-
vert would, by foriHula (3.) have been 3-3564* feet. The actual mean
daily motion of the united glaciers, between the lat and the S8th July, was
at Montanv6rt,+

" On the lat of July the centre of the aiilual motion of tlie Mer de Glao* a
Montanvert was 2-25 feet.

f Porbes' " Alps of Savoy," p. 140.

, Google

DO^ ArPEXDIX.

Neai' tlie side of tte gkuier - - 1'441 foet.
Between the side aud the centre - I'TBO "
Near tie ccnti'e ------ 2' 141 "

The motion of tha Glaoier de Leohant was therefore aooelerated by theii
confluence, and that of the Glacier du Geant retarded. The former ia
draped down hj the latter.

I have had tlie leas hesitatioa in offering this solution of the mechanical
problem of the motion of gladei's, m those hitherto proposed are con-
fessedly imperfect. That of De Sanssure, which attributes the descent of
the glacier simply to il3 weight, Is contradicted by the tkot that isolated
fragments of the glacier stand firmly on the slope on which the whole
nevertheless descends. It being obvious that if the parts would remain at
rest separately on the bed of the glacier, they would also remain at rest
when united.

That of Professor J. Forbes, which supposes a visoous or semi-flnid
structure of the glacier, is not consistent with the fact that no viscosity is
to be traced in its parts when separated. They appear as solid ihigments,
and they cannot acquire in their union properties in this respect which
individually they have not.

Lastly, the theory of Oharpentier, which attributes the descent of tlie
glacier to the daily congelation of the water which pei'colates it, and the
expansion of its mass consequent thereon, whilst it assigns a cause which,
so far as it operates, cannot, as I have shown, hut cause the glacier to
descend, appears to assign one inadequate to the result ; for the congelation
of the water which percolates the glacier does not, aocoi'ding to the obser-
vations of Professor Forbes,* take place at all in summer move than a few
inches from the surface. iTevei'theless, it Is in the summer that the ddly
motion of the glacier is the greateat.

The following remartable experiment of S£r. Hopkins of Oainbridge,t
which is considered by him to V)e confirmatory of the sliding theory of
De Sanssure as opposed to De Oharpentier's dilatation theory, receives
a ready esplanalion on the principles which I have Md down in this
note. It ia indeed a neceasaiy result of them. Mr. Hopkins placed a
mass of rough ice, confined by a square frame or bottomless bos, upon
a roughly chiselled fiag-stone, which he tlien inclined at a pmall angle;
and found tiat a slow but uniform motion was produced, when even it
was placed at an inconsiderable slope. This motion, which Mr. Hopkins
attiihttted to the dissolution of the ice in contact with the stone, would,
I apprehend, have taken place if the mass had been of lead instead of ice ;

* " Travels in the Alps," p. 41 3.

I I have quoted the foUowiiig account of it from Professor Forbes's book,
p. 419.

./Google

and it would have been but about balf as fast, because tba linear expan^
Bion of load, ia only about half that of ice.

The best Dimbn8ioh8 of a BcTTiiEffl.

Ie «J| (Art. 399.) represent the modulns of stabihty of the portion AG c(
the wall, it maj be shown, as before, tliat

T{{h,-~h)^m.<, — (l-a,—m„)co^.0.} = {ia, — m,){h, — h)a,f.;

. ■ . P{{S, — ft,)9in. o. — {l — a,)oo3. a}

=:i{ft,_S,)a,'ji_m,{Pco3.a + (S, — AJiii^}

If OTi=:™., the stability of the portion AU of the structure is the same
■witb that of tlie vihole AO ; an arrangeraunt by which the greatest
strength is obtained with a giyen quantity of material (see Art. 888.).
This supposition being made, and m eliminated between the above equa-
tion and equation (383,), that I'elation between the dimensions of the
buttress and those of the wall which is oonsistent with the greatest
economy of the material used will be determined. The following is that
relation ; —

i^(ffii'ft, + a(iiff^i + ■ a^h^ ~— V (S| sin. a, — I cos. a)

^ IMV-Wa,' -P{(h,-h,)mn. a-(^-a,)cos.cc}
,Pcos.o+/i»,(A,— Ss)

It is neoessftty to the greatest economy of the material of the Gothic
buttress (Art 801.) that the stability of the portions Qa and Q6, upon
their respective bases ae and be, should be same with that of the whole
buttress on its base EO. If, jo the preceding equation, Ai — S, be
substituted for h„ and h, — h, for A,, the resulting equation, together
with that deduced as esplained in the conclusion of Art. 301., will detei--
niine this condition, and ivill establish those relations between the dimen-
sions of the several portions of the buttress which are consistent with the
greatest economy of the material, or which yield the greatest sti'engtb to
the structure tcora the use of a ff.vea quantity of material.

, Google

DiMEsaiOKs 0

E Teets oe "WaHELa.

The foDowing rules are extracted from tlie work of !M. Morin, entitled
Aide Memo'ire de Mkard^ue Pratique : — If "we represent hy a the width
in parts of a foot of tlie tooth measi^red pai'allel to the axia of the wheel,
and by B its breadth or thickness measiirei parallel to the plane oi
rotation upon the pitch circle; then, the teeth heing constaJitly gi'eased,
the relation of a and 6 should be expressed, when the velocity of the pitch
circle does not esceed 5 feet per second, by a = 46 ; when it exceeds
6 feet per second, by a = 56 ; if the wheels are constantly exposed to wet,
by a = 66.

Thf^e relations being estabUahed, the ividth or tliicknees of tlie tooth
■wiU be deteniiined by the formnte contained in the columns of the follow-
ing table: —

M-.U.

'*"" u™d"kil?

^"^''td pmnds!' "^^

Cast iron ■
Brass
Hard wood

b =-10S V'P
6=143 1/?

6 = 002319 I'T
5 =-0028941/1'
i = 003iOS^P

Asanming that when the teeth are carefully executed the space between
the teeth should be y'jtli greater than their thickness, and y'^th greater
when the least labor is bestowed on them, the values of the pitch T will
in these two oases be represented by BCS+y'^) and i(,^+Ys), or by a-OB76
and 2'16. Substituting in these expressions the values of 6 ^ven by tlie
formnlse of the preceding table, then determining from the resulting
valnes of o (see equation 233.) the coiTesponding valnes of the coefB.cient
C (see equation 234.), the following table is obtained; —

„.«.

Talneofe(e^uaUoDE3S.).

Talus of C (equntlon SSi.). ]

the best work-
manship.

° m»^^*'

th°be^f«o*-
maoshlp.

lnferiofwc-rit-

Cast iron - -
Brass - -
Hard wood

■004785
■0066^1

■004870
■006077
■006726

0'M2
1-057
1-131

o^aaa

1-068
M4B

, Google

( OP CAEEIAGES. 685

The following are the pitches commonly in nae among mecLnnios : —

in. in. In. in. in. In. In.

1, Ih IS-, 1^, 2, 2i, 3.

Prof. "Willis considers the following to be sufQuient below ineli pitch : —

te. in. In. in. in.

i, f, h f, *■

Having, therefore, determined the proper pitch to he gi?ea to the tooth

from formala 284., the nearest pitch is to be tnlsea from the above series

to that thus determined.

EKPBRIMENT9 OF M. MOEIN ON THE ThAOTION OF OAlimAGES.

The following are among the general results dodacod bj M. Morin from
bis experimeBta : —

t. The traction is directly proportional to the load, and inversely pro-
portional to the diameter of tbe wheel.

3. Upon a paved or a hard Mscadamised rofwl, the resistance is independ-
ent of the width of the tire when it eireeeda from 8 to 4 inches.

8. At a walliing pace the traction is the same, mider the same circum-
stances, for carriages ■with springs and without them.

4. TTpon hard Macadamised and upon paved roads the traction increases
■with the velocity ; the increments of traetion being directly proportional
to the increments of velocity above the velocity 3^28 feet per aeoond, or
about 31 miles per hour. The equal increment of traction thus due to
each equal increment of velocity is less as the road is more smooth, and
the carriage less ri^d or better Lung

4. Upon soft roads of earth, or sand or turf, oi roads fresh and thioUy

graveled, the traction is independent of the velocity
6. Upon a well-made and compact p^vem6nt of hewn stones the traction

at a walking pace is not more than three-fourths of that upon the best

Macadamised road under similar circumstances, at a trotting pace it

is equal to it.
6. The destruction of the road in in all cnsea greater ai the dumetera

of the wheels are less, and it is jTieatT in ciiriagea without thin ■with

, Google

KOTE E.

Otf THE Stbehgte op CoiKMxa,

Me. Eoiigkinsos has obligingly communicated tte following observutious
on Art. 4B0. :—

1. The reader anist be made to tmderatand that the ronnding of the
ends of the pillars is to make them moveable there, as if they turned by
means of a nniversa! joint ; and the flat-ended pillura are conceived to be
supported in every pai-t of the ends by means of flat surfaces, or otherwieo
rendering the ends perfectly immoveable,

3. The coefficient (IS) for hollow oolnmns with ronnded ends is deduced
Itom the whole of the esperiments firat made, including some wliich were
very defeelive on account of the difficulty experienced in the earlier
attempt to oast good, hollow columns so small as were wanted. The
first castings were made lying on their side ; and this, notwithstanding
every effort, prevented the core being in the middle; some of the colnmna
were reduced, too, in thickness, half way between the middle and tie
ends, and near to the ends, aad this alightly reduced the strength. These
causes of weakness existed much more among the pillars witii rounded
ends than those with flat ones ; they are alluded to in the paper (Art. 47.).
Had it not been for them, the coefficient (18) would, I conceive, have
been equal to that for solid pillws (or 14*fl).

8, The fact of long pillars with flat ends being about three times as
strong (IS thos« of the same dimensions with rounded ends is, I conceive,
well made out, in cast iron, wrought iron, and timber; you have, how-
ever, omitted it, being perhaps led to do it through the low value of the
ooeffident (13) above mentioned.

The same may be mentioned with respect to the near approach m
strength of long pillars with flat ends, and those of haK the length with
rounded ends. It may be said that the law of the 1-7 power of the lengtli
would nearly indicate the latter ; but this last, and the other powers 3 76
and 8'55, are only approximations, and not exactly constant, though
nearly so, and I do not know whether the other equal qnantitiea are not,
witli some slight mollifications, physical facts.

4. The strength of pillars of aimilwr form and of the same materials
varies as the r865 power, or near as the square of their like linear
dimensions, or as the area of their cross section.

, Google

COMPLETE EI.UFTIO FrN0TI0S3. bb I

TABLE I.
The Numerical Values incomplete EUiptio Sanctions of the first and
SHOOND Orders fui-Yahte» nf the Modulus 'k.eorresponding to each Degree
qf the Angle sin.— 'S.

ro707B
1'6709I

l-MSfl
l-673la
1 '57 611

1-57079
1 '67067
l'S7U81
1-56972

1S4180
1-33286
1-82884
1-81473
1-30568

1-39627

1-60167

i-8oau8

1-61046

1-658R8
1-66073
1-64765
1-54416

ii-10465
2-13002
a -156 51
2-18421

a-aisi9

3-34854
2-27687

1-28012
1 ■22058
I-31106

2-41964
3-40099
3-6040G

1-67006
l-f.777S
1-68676

1-48643
1-48029
1-47396
1-49716

1-46077
1 -46390
\ -44639

2-9025fi
2-97856
3-06172

1-74149
1-76216
1-76335
1-77478
118676

1-42476
■1-41707
1-40923
1-40125

3-50042
S-66185
3-83174

4-05-275
4-3386S
4-74271
fi-4S490

1-03378
1-0^784
1-03281

./Google

TsB Tables ov M. Gairldel.

TABLE II.

Showing the Angle of Supture 't qfari Arch whose ZoaiUng is of the same
Material with its Voussoirs, and whose Sxtradoa is inclined at a ginen
Angle to the Eorison. (See Art. 844,)

a ^ ratio of lengths of voussoirs to radius of itiErados.

0 = i-atio of depth of load OTer crown to radius of intrados, so thiii

tf = 3(l+a). CArt.838.)

1 = inolhiatlon of extrados to horizon.

■

«.o

e^O-l

,....

fl=0-3

....

...

(=1-0

O'OB

680°

6919°

64-04'

6116°

49-86°

48-20°

45-74°

010

GS-4

60-48

67-70

56-01

M

^:^

54-17

52-34

0'15

64-0

61'3

B9-7

53-69

611

)

67-49

66-21

0'20

631

61-7

60-30

5^

10

58-80

0-26

61'76

61-22

16

60-59

0-30

61 '3

61-42

61-54

61-60

Ifi

61-8!

0-35

6017

60-80

61-21

61-64

61

JK

61-98

62-66

0'40

6S-8

69'8

60-62

61-05

HI

4N

61-67

62-9

0'45

51-32

68-S3

59-45

80-19

^o

61-23

62-85

0-60

5S'6li

56 97

68-09

58-98

59-72

60-34

62-40

^=0-1

e=0-»

...3

..M

o=m

^,

0 05

63-3°

67-3°

61-69°

48-61'

47-84°

46-ir

44-S5''

0-10

68

iH

55-96

'■>,

53

a

016

63-4B

fiil

17

68-33

ft;

',.',

67-00

66

II

65-66

61-48

611

4y

69-72

6V

16

59-05

6S

(7

68-29

60-75

flc

■ift

60-44

61

KN

60-SS

6(1

'.\

6017

W

+9

60-77

Kf

61-08

61

IR

61-48

60-62

61

n

61-38

61

63-31

61-18

^7

62-7

0-45

6711

bt

3n

69-29

60-06

60-67

61

16

6278

0-60

66-82

6718

58-21

69-08

59-81

60-41

62-46

, Google

AHG-Lli Oi' HUPrURl

-

c=^a

e=0'l

e=o-a

...<,

fcdl-5

^1.0

0-OS

M-B."

60-0°

4C-95°

45-69°

46-03°

44-67°

43-9°

69'8

6 5 '07

63-ai

!>'>.

61-99

1

Ift

fi908

57 'SS

58-66

fiK

15

55-75

55-66

66-05

■ (

M

fiS'Oa

58'aO

58-35

W

!.U

6810

68-03

5TB4

(

v.!'

S9-05

59'2B

59-42

5V

59-60

69-85

59-79

'

es-90

59-dT

59-98

61

•m

60-48

60-86

61-15

i.

S5

5B-S3

5941

60-09

6(

51

60-98

61-17

82-0

0-40

57-99

69-08

60-87

60-48

60-96

61-36

62-6

0-45

51 -an

68'48

69-84

60-06

60-67

61-16

62-7

O'BO

S6-88

51-81

58-58

59-S6

ao-oa

6084

62-B

-

„,

„.,

0=8-2

«=0«

c=i)-4

^.

0=1 -0

0-06

36-r

41-2°

43-0°

42-3°

42-6''

42-7°

42-9"

60-3

5019

50-17

50-14

50-18

50-11

0-15

64-81

54-S5

64-36

64-36

54-38

0"J0

66-n

se-ao

66-82

56-95

67-04

57-11

57-28

0-25

67-27

51-98

68-33

68-61

58-79

58-95

69-38

0-30

67-86

58-68

59-23

59-80

69-93

60-16

60 B8

0-35

58-07

69-01

59-70

60-ar

60-61

60-91

61-85

68-02

59-02

59-79

60-38

60 Bl

61-26

62-2

0-45

58-18

89-60

ao-26

eo-8-i

6127

62-7

0-50

61-30

58-31

59-16

59 88

60-41

61-00

62-9

•

.=0

..o-i

O.O.

„...

fl=8-l

c=0-5

fcil-0

0-05

313'

38-2°

38-4°

39-67°

40-38°

40-77°

41-9°

48-3

46-06

41-25

47-90

48 30

48-59

49-24

60 07

61-46

62-18

52-63

52-94

53-14

63-B8

64-69

65-21

65-67

65-96

56-16

66-12>

57-30

61-72

68-01

63-33

58-89

61-13

58-01

58-62

69-06

69-40

59-69

80-48

0-85

67-93

68-80

69-43

59-94

60-86

61-64

68-33

59-20

69-89

60-43

60-87

6289

58 41

60-03

60 61

61-08

6287

0-60

58-38

69-22

59-93

60-53

81-03

61-41

63-0

, Google

■

c=0

o=0-l

i!=0'a

..0-.

....

e=tli

w.

si-r

M-S"

36-38°

S7-fi9''

38-48=

3916*=

40-82°

0-10

40-98

4?

Sfl

46-67

47-14

48-86

0-15

4nii

41-

4(»

60-43

61-61

fil-96

52 -OS

0-20

52-01

fiV

ilK

04-01

M

-14

0-25

64-87

fifi

86-45

r.1

»4

67-39

67-09

63 '41

56-11

fi7

S8

53-16

fi^

w.

S8-98

69-26

80-16

nii

7H

EB-S4

r.(

*\

60-17

60-47

61-46

>fi

60-13

Ki

lid

60-97

61-30

es-2

ao-62

61

17

61-47

61-83

63-0

0-50

56-89

60-29

60-84

61-ea

61-72

62-07

68-3

"

,;=0

0=0-1

«=0-3

„.,

„...

0=0-6

0.1-0

0-OS

31-3°

38-68''

35-46°

B6-36''

87-22°

S8-0°

S9-B'

0-10

40-6

42-4

48-7

45-35

45-92

47-45

46-77

48'20

4il

IM

49-98

60-47

50-92

62-15

52-27

fi8

16

53-64

64-07

64-42

65-47

55-22

(4

56-31

66-70

5'i-Ol

57-97

0-30

E8-73

57-38

57

m

68-ao

68-65

68-94

59-85

68-35

E8-94

fill

4(1

6011

60-88

61 -SO

69-68

6009

6(

>2

60-89

61-19

61-46

62-4

60-40

60-89

61

61-67

61-97

62-24

63 2

0-50

60-99

61-43

61-8

62-2

62-5

62-8

, Google

EOEIZriNTAL THKUST OF AH AUCIi. 691

The Tables of M. Gabiuel.

TABLE III.

Showing tJw Horizontal Thrust of an Arch, the Sadius ofwhoge Introdas
i» JJaity, and the tcdght of each OuMo Foot o/its Material and that of
its Loadmg, Unity. (See Art. S44.)

IT.B. To find tlie horizontal thrust of any other arch, muhiply that given
in the table by the square of the radius of tlie intrados and by the weighi
of a cuHo foot of tie material.

o 0

0 O-l

0 0-4

e 0-8

-

P

P

P

ri

-w

f*

^

^

0-21763

0-28877

0-86060

0-48377

0-79541

0-10

0-10278

0-16370

0-22B88

0-28362

0'36ia4

0-41481

0-73161

0-16

0'118fl4

O-174B0

0-23111

0-2S7a4

0-34429

0-40100

0-68504

0-20

0-13013

O'lBlSl

0-23322

0-28460

0-33608

0-88747

0-64488

O'lBSTl

0-18658

0-2S287

0-21923

0-S26O7

0-37298

0-607 2T

0-30

0'X4333

0-1S604

0-2G874

0'3714B

0-81416

0-35687

0-57041

0'3S

O'I4504

0-18379

0-23258

0-26140

0-80028

0-33907

0-68385

0-40

0'I4422

0-17918

o-auiB

0-24924

0-28487

0-81953

0-49590

0-t4124

0-17240

0-20374

0-23630

0-26674

0-3988S

0-45693

0'1S64H

016396

O-10I6S

0-21S57

0-24780

0-27573

0-41728

o-O-S

e-O-S

e-0-4

«-fl-6

0 10

r'

^

rs

0-05

0-06180

0-12807

0-19937

0-27125

0-34366

0-41608

0-77 044

0-0 as 14

0-14666

0-20930

0-27237

0-38561

0-89895

0-71618

0-lB

0-10880

0-16001

0-2)667

0-37326

0-33003

0-38683

0-67110

0-20

0-1181B

0-16948

0-230B9

0-27237

0-82384

0*87 688

0-68286

0-26

0-12B70

0-17667

0-22244

0-26932

0-31819

0-36306

0-5974S

0 30

0-18698

0-17866

0-33134

0-38408

0-30673

0-34943

0-66295

0-140411

017909

0-21788

0-26661

0-29642

0-83424

0-52846

0-14234

0-17718

0-31215

0-24720

0-28230

0-31744

0-49844

0-45

0-14211

0-17823

02046^

0-33698

0-26761

0-39910

0-60

0-14008' 0-16753

1

0-19528

0-22319

0-25134| 0-27938

0-43096

, Google

(-0

0-0-1

o-o-a

C-0-&

O-0-4

o-O-B

0=1-0

r1

ra

■W

^

.^

■w

r!

O'oe

O'oesio

0-12266

0-194S8

0-26748

0-34018

0-41293

0-77681

010

0-07 BOS

0'14no

0-20493

0-26832

0-33176

0-89634

0-71377

O'lB

009990

0-X56fi8

0-21S36

O-2'7022

0-82708

0-88395

0-66840

0-20

0-11631

o-ieisi

0-21931

0-37 088

0-82284

0-37386

0-6814S

o-ae

0-13994

0-17582

0-22268

0-2695S

0-31648

0-86330

0-59767

0-30

0-13836

0-18096

0-23861

0-26627

0-80896

0-36168

0-56610

0'86

0'14494

0-183BS

0-22224

0-26096

0-29976

0-3886B

0-63271

0-40

0-1490S

0-18SB4

021878

0-26SS0

0-33399

0-4999S

o-4a

0-1S091

0-18212

0-21944

0-24488

0-37841

0-30800

0 48653

O'ao

0-16099

0-17860

0-20843

0-33439

0-26-247

0-29065

0-4323i

o-O

c-0-4

e-0-0

P

ri

!«

i«

r-

f

'"-

0-05

0-06102

0-13349

0-20621

0-27899

0-S6178

0-43468

078857

0-10

O-087O0

0-15053

0-21407

0-277 BO

0-34118

0-41 '466

07i3S3

0-15

0-10877

0-16567

0-23267

0-37947

088638

0-39328

0-67778

0-12685

0-17786

0-22986

0-28087

0-33289

0-88891

0-64150

0-2S

0-14037

0- 18716

0-23399

0-28082

0-32767

0-37463

0-60886

0-16139

0-19381

0-28640

0-37903

0-32166

0-86432

0-5777S

0-35

0-16943

0-19804

0-28669

0-27 640

0-S1415

0-36293

0-54700

0-40

0-1BS25

0-30005

0-23497

0-2B999

0-30506

0-S40I7

0-61608

0-45

0-20006

0-23141

0-26289

0-29444

0-32604

0-48460

0-50

017047

0-19824

0-32617

0-25433

0-28338

0-31060

0-45241

■

.=0

'r

T

'T

0-16
0-20
0-25
0-SO
0-35
0-40
0-46
0-50

0-09366
0-1)297
0-18295
0-15088
0-18498
0-17673
0-18699
0-19298
0-19774
0-20060

0-1B4O3
0-17692
0-18963
0-20173
0-21160
0-21917
0-22452
0-22777
0-22906
0-2-2864

0-28605
0-2K932
0-24a40
0-26314
0-26834
0-26170

0-36271
0-26060
0-2566!

0-80263
0-R0S2B
0-30459
0-80513
0-G0437
0-30182
0-29778
0-29ail2
0-28476

0-38101
0-86609
0-86009
0-35K06
0-36!9S
0-84688
0-34055
0-33-280
0-32861
0-31299

0-45365
0-42957
0-41696
0-40755
0-39876
0-88951
0-37980
0-36791
0-86624
0-34128

0-817S1
0-74711
0-701X8
0-66306
0-6H299
0-60282
0-57383
0-543H0
0-51385
0-48S-27

, Google

HOHIZONTAL THEUBT OF ^

,=3T° 30'.

c-o

"■

r!

r2

r^

r-

'"

O'OS

0-28864

0-36038

0-43 25 6

0-60490

0-86784

Old

0'15M9

0-221T4

0-28487

0-34768

0-41093

0-47426

0-19141

0-15

onaoB

0-a3iS3

0-28880

0-34653

0-40226

0-45904

9-74322

0-ao

0-16209

0'348ai

0-29448

0-84583

0-89722

0-44866

070598

025

0'20627

0-26282

0-2flB48

0-34619

0-39294

0-43972

067883

0-SO

0-36066

0-80814

0-84B6B

0-88826

0-4308S

0-64406

0-30S21

0-34388

0-B8359

0 42133

0-61529

0-40

oiMoea

0-37671

0-41088

0-58673

0 24130

0-272'; 6

0-30437

0-83586

0-36749

0-89918

0-55787

0'60

0'24499

0-27312

0-30182

0-32958

0-36789

0-38825

0-62845

0=^-1

P

P

^

rf

ra

^-

^

r!

003

0-23105

0-3O081

0-37162

0-44305

0-51485

0-68688

0-94381

0-10

0-23318

0-39S07

0-35764

0-42034

0-48333

0-34648

0-86300

0-15

0-24478

0-80079

0-BB7O8

0-41366

0-47013

0-62678

0-81059

0-20

0-25819

0-30915

0-36028

0-41151

0-46281

0-61416

0-77124

0-25

0-27104

0-31752

0-36410

0-41074

0-45744

0 50417

0-73809

0-30

0-28348

0-82486

0-86781

0-40981

0 45285

0-49493

0-70803

0-36

0-29216

0-33073

0-36935

0-40803

0-44674

0-48847

0-67939

0-40

0-39997

0-33494

0-86998

0-40606

0-44016

0-47530

0-65128

0-46

0-S0SB9

0-33746

0-86907

0-40072

0-43240

0-46412

0-62294

0-BO

0-30996

0-8S824

0-36667

0-39494

0-42334

0-45177

0-50419

, Google

ArPESDIX.

TABLE IV,
MecTidUKal Properties oftlie Materials <jf Oonstrvation.

Jfaie.~The Dspltala iffixed to 'Cos nombers in
B B VI S p rt haOm rniedami-s of

his table ret^r to tbe folloiriDg antlunmei:
La. Lam&

M, M\aabeabia<ii,Tntn>d.aiPlia.2^at.l
Ml. Mltla.
MtMualiot.

E. ^aaie\at,rXkdeBaUr,lY.

Eo. Eonnle, PM, TS'Ons. din.

T. Thomson, Oheoiiatn).

Te, TelfoM.

Tr. Tredgoid, JEsSBjr ok Va Slrett{;(!i qf

K

AoM E fr gr w1

B

448T

lOOQO Be.

1162000 B,

11202 B.

Air atm h

O0 828

o-oree

Alsiaate Mil

D ri tal white

d

noM.

141Sn M.

Bt M.

2S1'2S

1066 M,

Ash

g*

B3'ei

^ITSOTB.

isesBH.

ifl86SIL

}l6M30O B.

12166 B.

Bay-tree . . .

T16SH,

1C80

MOlflOO B,

20890 B.

Beech . . j

68 8T

SI

6102drylt

Blroh (common)

IfliB.

4»B0

ISOOO j

1562400 B.

lOOMB.

Do, (AnicrloiinJ .

■MSB.

40'Bri

Bismuth (Mat) . .

9-810 M.

618 81

8250 M,

IWiOM.

10299 H,

SfiW

10804 Ee

Do. (wii^e-drawn) .

SBl-OO

Briolt (refl) . .

ll^Ee.

18S6(i

280

Do. (pale red) . .

562 Eo

BrLuk-ffotk , .

I'SOO

Bullet-lree IBarbloe).

1089 B.

M31

2810600 B,

15688 B,

SBOO

eSWBe.

Cedar (Cang'dlsnifreah

■Bc»a

6614 H,

DiK (seoSDned) . .

-768

Chflll! . . i

■T94

lo -ees

moo

h-K

88* Ee,

-801 E,

g3!S!S5Si.,:

■m Br.

■M7Mt

Do. (coke). . .

Do. (Alfreton) . .

■2BBML

Do. (Bntterly) . .

281 Mt.

T900

Do. (Welsh stone) \

■890 Mt

iWeUh slaty)

■409 Mt.

88 06

Do

■278 Mt

'809 Mt

iocke). .

l-6STMt

108-M

Do, {Slaty). . .

1-MS m ' 5019

, Google

PEOPEKXIES OF MiTEEIALS OF CONSTEUOTIOJT,

OobI (BonlaTOoneen) .
Do. {oofcc) . .

Do. (onka) .

Do. isaiflocashire)

Do. fBwaoson) ,

Ocmpar (oast) .

ifo. (sheet) .

Do. (wiredrBWlO

Do, (In bolts) .
Ornb-trse , .'

Deal [Chriallaiia toM-

_. iemai middle)!
Do. ^ovway BpnicaJ
Do. (Kn^tsb) ,

EutH {rammad)
Elder , . .
Elm (seasoDed) .

Fir (Sew England)

Do. (Eiga) .
10, (Mat Forest)
'lint . . .

Olass (platt
Grarel

te (Aberdeen)

tComlah) ,

(red Egyptian)

0-696 M.
OCSSC.

Holly ;

Iron (wrouglt Euff ) ,
Do. (in bars) .
Do. Ibammared)

llnks,61noteL

Iron ^ Inch dlam. ,
Do. (Braoton's) with

251 tons, La.
KJ tons, la,
8U tons,T)ra,!
3T bins, La, :

e99S4()B.

2191200 B.
1836800 B,
699000 B.

10S0W H. ISOBSDOO H,

./Google

APPENDIX.

Do (BuffBiT, No.

sold blast)
"o. Ihot blast) .

o. (Coed Tidoii, 1

S, cold blast) .
Do. {bot bloat) .
Do. (F.lsloar, Nt

oolJ blast)
~- . (Miltofl, .

Du^lMdrkirk,

cold blast)
Do. (hotbluEt)

IflMF.
TfiMF.

T-296IL
T'22S H.

1070 n.
emu.

fl'BBS F.
8-B68 S.

T-IMP.
S'OTO P.

T-080P.

18-626
10500 Be.

, ITSJSiOOP. 4

'. 48M1P*
f. tdlOOF.*

r. sieeap."

SerS'.

Do.(wiral- .
Lignum TltB [oui
LlmMtona {iteiiM.
Do, (foliated) .
Do. (wlilte fluor)
Do. (gteeo) .

llmB (quiet) !

Mnhofflmj' (SpaniBli)
Maple ISoTWBY)
Marble (wbita IUlla_,
Do.ibVaokGolwav).
Metomy (at 82-)

Do. (DMit^o)

Do. (AiirlflOc) . .

Do. (AAlcim middle).

Do, jAniBr. jellow) .

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